Explanation:
Okay, so there's this problem about a hot-air balloon with ropes attached to four points around a central point O. The balloon is directly above O, and there are points A, B, C, D arranged around O: A is north, B is west, C is south, D is east. The distance between C and D is 140 meters. The ropes HA, HB, HC, HD are all connected from the balloon H to these points. Now, they want to replace ropes HC and HD with a single rope HP, where P is on the straight line between C and D. The goal is to figure out the greatest length of rope that can be saved by doing this replacement.

First, let me visualize this. Since A is north of O, B west, C south, D east, so O is the center. So if I imagine a coordinate system with O at (0,0), then A would be (0, y), B (-x, 0), C (0, -z), D (w, 0). But maybe coordinates can help here. The Asymptote code given probably has coordinates, but since it's not directly helpful, maybe I need to assign coordinates to figure out the positions.

The distance between C and D is 140 meters. If C is directly south of O, then coordinates of C would be (0, -c), and D is directly east of O, so (d, 0). Then the distance between C and D would be sqrt((d - 0)^2 + (0 - (-c))^2) = sqrt(d² + c²) = 140 m. So that's one equation.

Also, rope HC is 150 m and rope HD is 130 m. Since H is directly above O, let's say H has coordinates (0, 0, h) if we consider O as (0,0,0). But since the problem is on a flat field, maybe we can just treat it as 2D, with H being at (0, 0, h), and the points A, B, C, D being on the ground. Then the lengths of the ropes HA, HB, HC, HD are the distances from H to each of those points.

So if we consider H at (0, 0, h), then HC would be the distance from (0, 0, h) to C (which is, say, (0, -c, 0) if C is directly south of O. Wait, but coordinates can be assigned in such a way. Let me try to formalize this.

Let me set up coordinates. Let me assume O is at (0, 0). Let me define coordinates for A, B, C, D:

- A is directly north of O: so coordinates (0, a) for some a.

- B is directly west of O: coordinates (-b, 0) for some b.

- C is directly south of O: (0, -c) for some c.

- D is directly east of O: (d, 0) for some d.

Now, the distance between C and D is given as 140 m. So the distance between C(0, -c) and D(d, 0) is sqrt[(d - 0)^2 + (0 - (-c))^2] = sqrt(d² + c²) = 140. So we have that equation.

Now, the ropes from H to the points. Since H is directly above O, H must be at (0, 0, h) for some height h. Then the length of HC is the distance from H(0,0,h) to C(0, -c, 0), which is sqrt[(0 - 0)^2 + (-c - 0)^2 + (0 - h)^2] = sqrt(c² + h²) = 150 m. Similarly, HD is the distance from H to D(d, 0, 0): sqrt[(d - 0)^2 + (0 - 0)^2 + (0 - h)^2] = sqrt(d² + h²) = 130 m.

So we have two equations:

1. sqrt(c² + h²) = 150 → c² + h² = 150² = 22500.

2. sqrt(d² + h²) = 130 → d² + h² = 130² = 16900.

Also, from the distance between C and D:

sqrt(d² + c²) = 140 → d² + c² = 140² = 19600.

Wait, hold on! Wait, if sqrt(d² + c²) = 140, then d² + c² = 19600. But we also have from equations 1 and 2:

From equation 1: c² + h² = 22500.

From equation 2: d² + h² = 16900.

If we subtract the second equation from the first, we get (c² + h²) - (d² + h²) = 22500 - 16900 → c² - d² = 5600.

But from the distance between C and D: d² + c² = 19600.

So now we have:

c² - d² = 5600,

c² + d² = 19600.

If we add these two equations: 2c² = 5600 + 19600 = 25200 → c² = 12600 → c = sqrt(12600) = sqrt(100 * 126) = 10*sqrt(126) = 10*sqrt(9*14) = 10*3*sqrt(14) = 30√14 ≈ ... but maybe exact value is not needed yet.

Similarly, subtracting the first equation from the second: (c² + d²) - (c² - d²) = 19600 - 5600 → 2d² = 14000 → d² = 7000 → d = sqrt(7000) = sqrt(100*70) = 10*sqrt(70).

So now we can find h² from equation 2: d² + h² = 16900 → h² = 16900 - d² = 16900 - 7000 = 9900 → h = sqrt(9900) = sqrt(100*99) = 10*sqrt(99) = 10*sqrt(9*11) = 30√11. Wait, sqrt(99) is 3√11? Wait, sqrt(99) is sqrt(9*11) which is 3√11. Yes. So h is 30√11 meters.

So now, we know c, d, h. But maybe we don't need the exact values? Wait, maybe we need coordinates for P?

But the problem is about replacing ropes HC and HD with a single rope HP where P is on CD. So the original total length of ropes HC and HD is 150 + 130 = 280 m. The new length would be HP, so the amount saved is 280 - HP. We need to find the maximum possible value of this, which is equivalent to minimizing HP. Therefore, the greatest length that can be saved is 280 - min(HP). Therefore, we need to find the minimal HP where P is on segment CD.

Therefore, the problem reduces to: find the minimal distance from H to the line segment CD, and then subtract that from 280.

But wait, hold on. However, since H is above O, and CD is a line segment between C and D. So HP is the straight line from H to P, where P is somewhere on CD. Therefore, the minimal HP is the minimal distance from H to the line CD, provided that the foot of the perpendicular from H to CD lies on the segment CD. If it does, then that's the minimal distance. Otherwise, the minimal distance would be the distance to the nearest endpoint. Therefore, first we need to check if the perpendicular from H to CD intersects CD. If so, then the minimal HP is that distance. If not, then it's the distance to C or D.

So let's find coordinates of H, C, D, then find the minimal distance from H to CD.

Earlier, we had coordinates for C(0, -c), D(d, 0), and H is at (0, 0, h). Wait, but in the coordinate system, if we are working in 2D, then H is at (0, 0) with height h? Wait, actually, since we are dealing with distances in 3D (since H is above the ground), but the points A, B, C, D are on the ground. So the coordinates for C and D are (0, -c, 0) and (d, 0, 0), and H is (0, 0, h). Then the line CD is from (0, -c, 0) to (d, 0, 0). The point P is somewhere on that line. The distance from H to P would be the length of HP, which is sqrt( (x_P - 0)^2 + (y_P - 0)^2 + (0 - h)^2 ) where (x_P, y_P, 0) is a point on CD.

Alternatively, since P lies on CD, we can parametrize CD. Let me parametrize CD. Let me consider CD as a line segment between C(0, -c, 0) and D(d, 0, 0). Let me parametrize P as follows: for t in [0, 1], P(t) = C + t*(D - C) = (0 + t*d, -c + t*(0 - (-c)), 0) = (td, -c + tc, 0) = (td, -c(1 - t), 0).

Therefore, coordinates of P(t) are (td, -c(1 - t), 0).

Then, the distance HP is sqrt( (td - 0)^2 + (-c(1 - t) - 0)^2 + (0 - h)^2 ) = sqrt( t²d² + c²(1 - t)² + h² ).

Therefore, HP² = t²d² + c²(1 - t)² + h². To minimize HP, we can minimize HP².

So let's consider f(t) = t²d² + c²(1 - t)^2 + h².

Let me compute f(t):

f(t) = d² t² + c² (1 - 2t + t²) + h² = (d² + c²) t² - 2c² t + c² + h².

But we already know that d² + c² = 19600. Therefore, f(t) = 19600 t² - 2c² t + c² + h².

So to find the minimum of this quadratic function in t. Since it's a quadratic in t, the minimum occurs at t = [2c²]/[2*19600] = c² / 19600.

Wait, more precisely, for a quadratic function f(t) = at² + bt + c, the vertex is at t = -b/(2a).

In our case, f(t) = 19600 t² - 2c² t + (c² + h²). Therefore, a = 19600, b = -2c². So the minimum occurs at t = (2c²)/(2*19600) = c² / 19600.

Therefore, t_min = c² / 19600.

But we need to check if t_min is within [0, 1], since t is between 0 and 1 for the segment CD.

So let's compute t_min:

We have c² = 12600 (from earlier: c² = 12600). Therefore, t_min = 12600 / 19600 = 126 / 196 = 9/14 ≈ 0.6428... which is between 0 and 1. Therefore, the foot of the perpendicular from H to CD lies on the segment CD. Therefore, the minimal HP is achieved at t = 9/14.

Therefore, the minimal HP is sqrt(f(t_min)).

Alternatively, since we can compute f(t_min):

f(t_min) = 19600*(c² / 19600)^2 - 2c²*(c² / 19600) + c² + h².

Wait, maybe there is a better way. Since we know that H is at (0, 0, h), and P lies on CD, then the minimal distance from H to CD is the length of the perpendicular from H to line CD. However, since we have coordinates, maybe we can compute that.

Alternatively, since we have already parametrized it, maybe we can compute f(t_min). Let me try another approach.

Alternatively, since we have coordinates of H, C, D, we can find the equation of line CD and then compute the distance from H to line CD.

But since H is not on the same plane as CD? Wait, CD is on the ground (z=0), and H is at (0, 0, h). So the distance from H to line CD is the minimal distance between H and any point on CD. Which is the same as sqrt( (distance from H to projection on plane)^2 + (height)^2 )? Wait, no, since the line CD is on the plane z=0, and H is at (0,0,h). The minimal distance from H to line CD is the minimal distance between H and points on CD. Which is the same as the minimal sqrt( (x)^2 + (y)^2 + h² ) where (x, y) is on line CD.

But since (x, y) is on CD, which goes from C(0, -c) to D(d, 0). So parametrizing CD as earlier: (td, -c(1 - t)).

Therefore, distance squared is (td)^2 + (-c(1 - t))^2 + h², same as before. So yes, so the minimal value is f(t_min) as above.

But maybe we can compute it more easily. Let me recall that in 3D, the distance from a point to a line can be calculated by the formula:

Distance = |(P0 - P) × v| / |v|,

where P is the point, P0 is a point on the line, and v is the direction vector of the line.

So let's try that. Let me take P as H(0, 0, h). Let me take P0 as point C(0, -c, 0). The direction vector of CD is D - C = (d, 0, 0) - (0, -c, 0) = (d, c, 0). So direction vector v = (d, c, 0).

Then vector P0 - P is C - H = (0 - 0, -c - 0, 0 - h) = (0, -c, -h).

Then the cross product (P0 - P) × v is determinant:

i   j   k

0  -c  -h

d   c   0

Calculating determinant:

i * [(-c)(0) - (-h)(c)] - j * [0*0 - (-h)*d] + k * [0*c - (-c)*d]

= i*(0 + hc) - j*(0 + hd) + k*(0 + cd)

= (hc, -hd, cd)

Then the magnitude of this cross product is sqrt( (hc)^2 + (-hd)^2 + (cd)^2 ) = sqrt( h²c² + h²d² + c²d² )

Then divide by |v|, which is sqrt(d² + c²) = 140.

So distance is sqrt( h²c² + h²d² + c²d² ) / 140.

Therefore, distance squared is [ h²c² + h²d² + c²d² ] / 140².

But wait, distance squared is (hc)^2 + (hd)^2 + (cd)^2 divided by 140²?

Wait, no. Wait, the distance is |(P0 - P) × v| / |v|. So squared distance is [ |(P0 - P) × v|² ] / |v|².

So |(P0 - P) × v|² is (hc)^2 + (hd)^2 + (cd)^2. Therefore, squared distance is [ h²c² + h²d² + c²d² ] / (d² + c²)².

But since d² + c² = 19600, so denominator is 19600².

But let me compute numerator:

h²c² + h²d² + c²d² = h²(c² + d²) + c²d².

From earlier, c² + d² = 19600, and c²d² is (cd)^2. Wait, but we can compute h²(c² + d²) + c²d². Let's compute:

We know c² + d² = 19600.

From equations 1 and 2:

From equation 1: c² + h² = 22500 → h² = 22500 - c².

Similarly, equation 2: d² + h² = 16900 → h² = 16900 - d².

But maybe another way. Let's compute h²(c² + d²) + c²d²:

= h²*(c² + d²) + c²d².

We have h² = ?

Earlier, we found h² = 9900. Wait, from equation 2: d² + h² = 16900. Since d² = 7000, then h² = 16900 - 7000 = 9900. Yes. So h² = 9900. Therefore:

h²*(c² + d²) + c²d² = 9900 * 19600 + c²d².

But we also can compute c²d². Since c² = 12600, d² = 7000, so c²d² = 12600 * 7000.

Therefore:

Numerator is 9900 * 19600 + 12600 * 7000.

Let me compute this. Let me note that 12600 = 126 * 100, 7000 = 70 * 100, 9900 = 99 * 100, 19600 = 196 * 100. So maybe factor out 100^2:

Numerator = (99 * 196 * 100^2) + (126 * 70 * 100^2) = 100^2 [99*196 + 126*70].

Compute 99*196:

Well, 100*196 = 19600, minus 1*196 = 19600 - 196 = 19404.

Then 126*70 = (120 + 6)*70 = 120*70 + 6*70 = 8400 + 420 = 8820.

So total inside the brackets is 19404 + 8820 = 28224.

Therefore, numerator is 100^2 * 28224. Therefore, squared distance is [100² * 28224] / [19600²] = [28224 / 19600²] * 100² = ?

Wait, let me calculate:

Squared distance = [Numerator] / [denominator] = [100² * 28224] / [19600²] = (28224) / (19600² / 100²) ) = 28224 / ( (19600 / 100)^2 ) = 28224 / (196²).

Compute 196²: 196*196. Well, 200² = 40000, minus 4*200 + 4 = 40000 - 800 + 4 = 39204? Wait, no. Wait, (a - b)^2 = a² - 2ab + b². Let me compute 196²:

196 * 196:

Well, 200 * 196 = 39200,

Minus 4 * 196 = 784,

So 39200 - 784 = 38416. Yes, 196² = 38416.

Therefore, squared distance = 28224 / 38416.

Simplify this fraction. Let's divide numerator and denominator by 16:

28224 / 16 = 1764,

38416 / 16 = 2401.

So 1764 / 2401.

Check if they can be simplified further. Let me check GCD of 1764 and 2401.

Factor 1764: 42² = (6*7)^2 = 6²*7² = 36*49. Also, 1764 is 49*36 = 49*36.

Factor 2401: 49*49 = 2401. So 2401 = 49². Therefore, 1764 = 49*36, 2401 = 49². Therefore, simplify 1764 / 2401 = (49*36)/(49*49) ) = 36 / 49.

Therefore, squared distance is 36/49. Therefore, distance is sqrt(36/49) = 6/7. Wait, but what are the units here? Because the numerator and denominator?

Wait, hold on. Wait, we computed squared distance as [Numerator]/[Denominator] = 28224 / 38416 = 36/49. So sqrt(36/49) = 6/7. But what are the units?

Wait, actually, we had:

Squared distance = [ |(P0 - P) × v|² ] / |v|² = [ h²c² + h²d² + c²d² ] / (d² + c²)^2 = 36/49.

But wait, that would mean the distance is 6/7? But that seems odd because all the lengths were in meters. Wait, no, because we had earlier calculated numerator as (h²c² + h²d² + c²d²) which is in units of meters squared. Wait, no, let me retrace.

Wait, when we computed the numerator, which was |(P0 - P) × v|² = h²c² + h²d² + c²d². But we computed that as 100² * 28224, but in reality, those numbers were in terms of the variables.

But actually, perhaps I made a miscalculation here. Because when I calculated:

Numerator = h²(c² + d²) + c²d².

But h² = 9900, c² + d² = 19600, and c²d² = 12600 * 7000.

So h²(c² + d²) = 9900 * 19600, and c²d² = 12600 * 7000.

But when we did the calculation, we considered the numerical values. Wait, but in reality, that expression is:

Numerator is h²(c² + d²) + c²d².

But we can compute this as:

= h²*(c² + d²) + c²d².

But since h² = 22500 - c², from equation 1:

Wait, equation 1 was c² + h² = 22500, so h² = 22500 - c². Therefore, substitute:

Numerator = (22500 - c²)(c² + d²) + c²d².

But c² + d² = 19600, so:

= (22500 - c²)(19600) + c²d².

But we also know that d² = 19600 - c². So:

= (22500 - c²)(19600) + c²(19600 - c²)

= 22500*19600 - c²*19600 + 19600 c² - c^4

= 22500*19600 - c^4.

So the numerator is 22500*19600 - c^4.

But c² = 12600, so c^4 = (12600)^2.

So numerator = 22500*19600 - (12600)^2.

Hmm, this seems more complicated. However, earlier calculation said that this equals 100² * 28224. Maybe there was a miscalculation.

Wait, but if we use the previous approach where we found that squared distance is 36/49, that would mean that the distance from H to line CD is 6/7 of some unit? But this seems contradictory.

Alternatively, maybe I messed up the cross product calculation? Let me check again:

Earlier, cross product (P0 - P) × v was (hc, -hd, cd). Then its magnitude squared is h²c² + h²d² + c²d². Then divided by |v|² which is (d² + c²). Therefore, squared distance is [h²(c² + d²) + c²d²]/(c² + d²)^2.

But that's equal to [h² + (c²d²)/(c² + d²)] / (c² + d²).

Wait, maybe not. Wait, more accurately, squared distance = [h²(c² + d²) + c²d²]/(c² + d²)^2 = [h²/(c² + d²)] + [c²d²/(c² + d²)^2].

But maybe we can compute this with our known values:

We have c² + d² = 19600, h² = 9900, c²d² = 12600 * 7000.

So let's compute h²(c² + d²) + c²d² = 9900 * 19600 + 12600 * 7000.

Let me compute 9900 * 19600:

Note that 9900 = 10000 - 100, so (10000 - 100)*19600 = 10000*19600 - 100*19600 = 196,000,000 - 1,960,000 = 194,040,000.

Then 12600 * 7000. Let's compute 12600 * 7000. 12600 * 7000 = 12.6 * 10^3 * 7 * 10^3 = 88.2 * 10^6 = 88,200,000.

Therefore, total numerator is 194,040,000 + 88,200,000 = 282,240,000.

Denominator is (c² + d²)^2 = (19600)^2 = 384,160,000.

Therefore, squared distance is 282,240,000 / 384,160,000 = 282,240 / 384,160 = divide numerator and denominator by 160: 282,240 ÷ 160 = 1764, 384,160 ÷ 160 = 2401. So 1764 / 2401 as before. Which reduces to 36 / 49. Therefore, squared distance is 36/49, so distance is 6/7 meters? Wait, no. Wait, the units here are?

Wait, the numerator was h²c² + h²d² + c²d², which has units of (m² * m²) = m⁴? Wait, no. Wait, h is in meters, c is in meters, so h²c² is (m²)(m²) = m⁴. Similarly, h²d² is m⁴, c²d² is m⁴. Then the numerator is m⁴, denominator is (m²)^2 = m⁴. Therefore, distance squared is dimensionless? Wait, no:

Wait, the distance is in meters. Wait, no. Wait, the cross product [(P0 - P) × v] has units of meters^3? Wait, original cross product: (P0 - P) is a vector from H to C, which is in meters in each component. v is direction vector of CD, also in meters. Therefore, cross product has units of meters^2. Then |(P0 - P) × v| has units meters². Divided by |v| (meters), so the distance has units meters. Therefore, squared distance has units meters². Therefore, our earlier calculation of 36/49 must be in units of meters squared? But how?

Wait, we did the computation with actual numbers:

Numerator was 282,240,000 m⁴? Wait, no. Wait, the actual calculation:

Wait, let me think again. Let me check with specific numbers. Let me think of all variables in meters:

h is in meters, c and d are in meters. Therefore, h²c² has units of meters^4. Therefore, the numerator [h²c² + h²d² + c²d²] has units of meters^4. The denominator (c² + d²)^2 is meters^4. Therefore, the ratio is dimensionless? But then how come we get a numerical value of 36/49?

Wait, there must be a miscalculation here. Because if the distance squared is 36/49, then that would mean the distance is 6/7 meters, but that can't be, since HC is 150 meters, which is much longer. Therefore, clearly, I messed up the calculation of the cross product.

Wait, where did the 36/49 come from? It seems the problem is that I forgot that the cross product is calculated in 3D space. Wait, in our case, the cross product vector was (hc, -hd, cd). Then its magnitude squared is (hc)^2 + (-hd)^2 + (cd)^2 = h²c² + h²d² + c²d². But then divided by |v|² which is (d² + c²). Therefore, the squared distance is [h²(c² + d²) + c²d²]/(c² + d²)^2.

But this is equal to [h²/(c² + d²)] + [c²d²/(c² + d²)^2].

So let's compute that with numbers:

h² = 9900, c² + d² = 19600, c²d² = 12600 * 7000 = 88,200,000.

So:

First term: h²/(c² + d²) = 9900 / 19600 ≈ 0.5051.

Second term: c²d²/(c² + d²)^2 = 88,200,000 / (19600)^2. Let me compute 19600² = 384,160,000. So 88,200,000 / 384,160,000 ≈ 0.2297.

Total ≈ 0.5051 + 0.2297 ≈ 0.7348. Which is approximately 0.7348, which is roughly 0.7348. What is 36/49? 36 divided by 49 is approximately 0.73469... Yes! So 0.7348 ≈ 36/49. So that's correct. Therefore, squared distance is 36/49, so distance is 6/7 meters? Wait, no! Wait, no! Wait, squared distance is 36/49, but that's in meters squared? No, no! Wait, the actual calculation is:

The distance squared is [h²(c² + d²) + c²d²]/(c² + d²)^2. But this is equal to (from previous) 36/49. However, this is a dimensionless quantity? No! Wait, no, the numerator and denominator both have units of meters⁴? No, no, actually, c² + d² is meters², so (c² + d²)^2 is meters⁴, and the numerator is h²(c² + d²) + c²d², which is (m² * meters²) + meters⁴ = meters⁴ + meters⁴ = meters⁴. Therefore, the ratio is (meters⁴) / (meters⁴) = dimensionless. But we have a numerical value of 36/49. Which means that the squared distance is 36/49*(some unit)? No, no, this is not correct. Wait, there must be an error in the calculation. How can the distance squared be dimensionless?

Wait, no! Wait, no, the formula is:

Distance squared = [ |(P0 - P) × v|² ] / |v|².

Which is [ (meters²)² ] / (meters²) ) = meters² / meters² = dimensionless? No! Wait, no. Wait, (P0 - P) is a vector in 3D space, with units of meters. v is also a vector with units of meters. The cross product (P0 - P) × v has units of meters × meters = meters². Then squared magnitude is meters⁴. Divided by |v|² which is meters². Therefore, the distance squared has units of meters². Therefore, the numerical value 36/49 is multiplied by (meters²)? Wait, no. Wait, if I compute [h²(c² + d²) + c²d²]/(c² + d²)^2, which is 36/49, then that is equal to distance squared? But how can that be?

Ah! Wait, no, there was a mistake in the previous cross product calculation. Wait, the cross product [(P0 - P) × v] has magnitude |(P0 - P) × v|, which is equal to |P0 - P| * |v| * sin(theta), where theta is the angle between them. Therefore, the squared magnitude is |P0 - P|² * |v|² * sin²(theta). Then, divided by |v|² gives |P0 - P|² * sin²(theta). Which is equal to the squared distance from H to line CD.

Wait, but in reality, the distance from H to line CD is equal to |(P0 - P) × v| / |v|. Therefore, the squared distance is |(P0 - P) × v|² / |v|².

But according to our calculation, this equals 36/49. However, this suggests that the squared distance is 36/49 in meters squared? But 36/49 is just a number. Therefore, we must have forgotten to multiply by something?

Wait, no. Wait, the calculation of [h²(c² + d²) + c²d²]/(c² + d²)^2 resulted in 36/49. But that 36/49 is actually [(h²(c² + d²) + c²d²)] / (c² + d²)^2. However, in reality, this must have units. But since we plugged in the actual numerical values, the result is a numerical value times meters squared? No, no. Wait, let me think again.

Wait, maybe there was a miscalculation earlier where I thought the numerator was 282,240,000 and denominator was 384,160,000. Wait, but these numbers were in terms of the actual variables. Wait, actually, if we compute:

Numerator: h²(c² + d²) + c²d² = 9900 * 19600 + 12600 * 7000.

We calculated that as 194,040,000 + 88,200,000 = 282,240,000.

Denominator: (c² + d²)^2 = 19600² = 384,160,000.

Therefore, distance squared is 282,240,000 / 384,160,000 = 0.73469... which is 36/49. Therefore, distance is sqrt(36/49) = 6/7. But what is the unit? Since all the terms in numerator and denominator have units of meters⁴? Wait, no, no. Wait, h² is meters², (c² + d²) is meters², so h²*(c² + d²) is meters⁴. Similarly, c²d² is meters⁴. Therefore, the numerator is meters⁴, denominator is (meters²)^2 = meters⁴. Therefore, the ratio is dimensionless, so distance squared is 36/49, meaning distance is 6/7 meters? That's not possible because the distance from H to line CD can't be less than the distance from H to C or H to D, which are 150 and 130 meters.

Wait, this must mean there is an error in my calculations. Because 6/7 meters is about 0.857 meters, which is way smaller than 130 meters. Therefore, there's a miscalculation.

Wait, but where? Let me check the cross product again.

The cross product (P0 - P) × v. P0 is C(0, -c, 0). P is H(0, 0, h). Therefore, vector P0 - P is C - H = (0 - 0, -c - 0, 0 - h) = (0, -c, -h).

Vector v is direction vector of CD: D - C = (d, c, 0).

So cross product is determinant:

i   j   k

0  -c  -h

d   c   0

Which is i*( (-c)(0) - (-h)(c) ) - j*( (0)(0) - (-h)(d) ) + k*( (0)(c) - (-c)(d) )

= i*(0 + hc) - j*(0 + hd) + k*(0 + cd)

= (hc, -hd, cd)

Yes, that's correct. Then the magnitude squared is h²c² + h²d² + c²d². Then divided by |v|² = d² + c². Therefore, the squared distance is [h²(c² + d²) + c²d²] / (c² + d²)^2.

But with actual numbers, we said that [h²(c² + d²) + c²d²] = 282,240,000 and (c² + d²)^2 = 384,160,000, giving 0.73469... which is 36/49.

But if this is supposed to be the squared distance, then distance is sqrt(36/49) = 6/7 meters. Which is impossible. Therefore, there must be a miscalculation in the numerical values.

Wait, let me check again the value of h²(c² + d²) + c²d²:

h² = 9900, c² + d² = 19600, c²d² = 12600*7000 = 88,200,000.

So h²*(c² + d²) = 9900 * 19600 = Let me compute this carefully.

First compute 9900 * 19600.

Note that 9900 * 19600 = (10000 - 100) * 19600 = 10000*19600 - 100*19600 = 196,000,000 - 1,960,000 = 194,040,000. Yes.

Then c²d² = 12600 * 7000. Let me compute that as 12600 * 7000.

Note that 12600 * 7000 = 12600 * 7 * 1000 = 88200 * 1000 = 88,200,000. Correct.

So total numerator is 194,040,000 + 88,200,000 = 282,240,000.

Denominator is (19600)^2 = 384,160,000.

So the ratio is 282,240,000 / 384,160,000 = dividing numerator and denominator by 1000: 282,240 / 384,160.

Divide numerator and denominator by 160: 282,240 ÷ 160 = 1764; 384,160 ÷ 160 = 2401.

So 1764 / 2401.

Now, 1764 is 42², and 2401 is 49². So 1764 / 2401 = (42/49)² = (6/7)² = 36/49.

Ah! Therefore, the squared distance is (6/7)². Wait, no, the squared distance is [h²(c² + d²) + c²d²]/(c² + d²)^2 = (6/7)^2. Wait, but this is the squared distance? Wait, no! Wait, no, the squared distance is [ |(P0 - P) × v|² ] / |v|² = (282,240,000) / (384,160,000) = 36/49. Therefore, distance is sqrt(36/49) = 6/7. But what are the units?

Wait, no! This is where the error is. The cross product [(P0 - P) × v] has magnitude sqrt(h²c² + h²d² + c²d²), which we calculated as sqrt(282,240,000). Wait, no. Wait, no. Wait, we had |(P0 - P) × v|² = 282,240,000. Then dividing by |v|² = (19600)^2 = 384,160,000 gives us (distance)^2 = 282,240,000 / 384,160,000 = 36/49. So distance = 6/7. But this is in what units?

Wait, the problem is that all the terms in the numerator and denominator are meters⁴, so dividing them gives a dimensionless quantity. Therefore, the numerical value 6/7 is in (m²)? No, no. Wait, actually, no. Wait, the formula is:

distance = |(P0 - P) × v| / |v|.

But |(P0 - P) × v| is sqrt(h²c² + h²d² + c²d²). Wait, but h, c, d are in meters. So h²c² has units of m⁴? No! Wait, h is in meters, c is in meters, so h²c² is meters² * meters² = meters⁴. Therefore, |(P0 - P) × v| has units of sqrt(m⁴) = meters². Then divide by |v| which is in meters, so overall distance has units of meters² / meters = meters. Therefore, the numerical value 6/7 is multiplied by meters?

Wait, I think I need to realize that in the calculation, we used specific values for h, c, d, but forgot to carry the units. Let me re-express this.

Let me assume that all lengths are in meters. Then h, c, d are in meters. Let me compute the numerator |(P0 - P) × v|² = h²c² + h²d² + c²d². This is in (m²)(m²) + (m²)(m²) + (m²)(m²) = m⁴.

Denominator |v|² is meters².

Therefore, the entire expression is [m⁴] / [m²] = m². Therefore, the distance squared is in m², so distance is in meters.

But how does the numerical value of 36/49 translate to actual meters? Because we calculated it as a ratio of numbers. So the numerical value is 36/49, but multiplied by what? Wait, we have to realize that the actual calculation of |(P0 - P) × v|² is equal to 282,240,000, which is meters⁴. Therefore, divided by |v|² = 384,160,000 meters², gives distance squared = 282,240,000 / 384,160,000 meters² = 0.73469... meters². Therefore, distance is sqrt(0.73469...) meters ≈ 0.857 meters. Wait, but why is this number?

Wait, but this contradicts the fact that HC is 150 meters. How can the distance from H to CD be only 0.857 meters? That doesn't make sense. So clearly, there is a miscalculation.

Wait, this suggests that I messed up the cross product. Maybe I confused the formula? Let me think again.

Wait, the formula for the distance from a point to a line in 3D is |(P0 - P) × v| / |v|, where P is the point, P0 is a point on the line, and v is the direction vector of the line.

But in our case, P is H(0, 0, h), P0 is C(0, -c, 0), v is D - C = (d, c, 0).

So the vector P0 - P is (0 - 0, -c - 0, 0 - h) = (0, -c, -h).

Then the cross product is:

(P0 - P) × v = |i   j   k|
               0  -c  -h
               d   c   0

= i * [(-c)(0) - (-h)(c)] - j * [0*0 - (-h)(d)] + k * [0*c - (-c)(d)]

= i * [0 + hc] - j * [0 + hd] + k * [0 + cd]

= (hc, -hd, cd)

The magnitude squared of this vector is (hc)^2 + (hd)^2 + (cd)^2 = h²c² + h²d² + c²d².

But this is supposed to be |(P0 - P) × v|².

Therefore, the distance from H to line CD is sqrt(h²c² + h²d² + c²d²) / sqrt(d² + c²).

Let me compute this expression with our numerical values.

First compute numerator: sqrt(h²c² + h²d² + c²d²).

We had h²c² + h²d² + c²d² = 282,240,000. Wait, no! Wait, h²c² + h²d² + c²d² is equal to 9900*12600 + 9900*7000 + 12600*7000? No, no! Wait, earlier we said h²c² + h²d² + c²d² = h²(c² + d²) + c²d² = 9900*19600 + 12600*7000 = 282,240,000. So that is in meters⁴. Therefore, sqrt(282,240,000) is in meters², and then divided by sqrt(19600²) = 19600 meters. Therefore, the distance is sqrt(282,240,000)/19600 meters.

Let me compute sqrt(282,240,000). Let's note that 282,240,000 = 282,240 * 1000. Wait, but sqrt(282,240,000) = sqrt(282,240 * 1000) = sqrt(282,240) * sqrt(1000). Wait, but this seems messy. Wait, but earlier we said that this is equal to 6/7 * something?

Wait, no. Let me instead recall that we had [h²(c² + d²) + c²d²]/(c² + d²)^2 = 36/49. Therefore, distance squared is 36/49. Wait, but that can't be. Wait, no! No, the distance squared is [h²(c² + d²) + c²d²]/(c² + d²)^2? No, the distance squared is [h²(c² + d²) + c²d²]/(c² + d²)^2? No, the distance squared is [ |(P0 - P) × v|² ] / |v|² = [h²(c² + d²) + c²d²]/(c² + d²)^2. Wait, but this is equal to 36/49? Which would mean that distance squared is 36/49, but I think the units are missing. Wait, no. If all the terms are in meters, then the [h²(c² + d²) + c²d²] is in meters⁴, and divided by (meters²)^2, which is meters⁴, giving a dimensionless quantity. But then, how can this be?

Wait, I think the error is that I assumed that the cross product's magnitude squared was equal to h²c² + h²d² + c²d², but in reality, no. Let me recast the calculation with actual numerical values. Let me take specific numbers for h, c, d.

Let me recall:

c² = 12600, d² = 7000, h² = 9900.

Therefore, c = sqrt(12600) ≈ 112.25 m, d = sqrt(7000) ≈ 83.666 m, h = sqrt(9900) ≈ 99.499 m.

So let's compute the numerator, h²c² + h²d² + c²d².

h²c² = 9900 * 12600 = let's compute 9900 * 12600.

Well, 10,000 * 12,600 = 126,000,000

Minus 100 * 12,600 = 1,260,000

So 126,000,000 - 1,260,000 = 124,740,000.

Similarly, h²d² = 9900 * 7000 = 69,300,000.

c²d² = 12,600 * 7,000 = 88,200,000.

So total numerator: 124,740,000 + 69,300,000 + 88,200,000 = 124,740,000 + 69,300,000 = 194,040,000 + 88,200,000 = 282,240,000.

Denominator: (c² + d²)^2 = (19600)^2 = 384,160,000.

So distance squared = 282,240,000 / 384,160,000 = 0.73469...

Now, distance = sqrt(0.73469) ≈ 0.857 meters. But how can this be? If H is at 99.5 meters above O, and C is 112.25 meters south of O, then the distance from H to C is sqrt( (0 - 0)^2 + (-c - 0)^2 + (0 - h)^2 ) = sqrt(c² + h²) = sqrt(12600 + 9900) = sqrt(22500) = 150 m. Similarly, distance from H to D is sqrt(d² + h²) = sqrt(7000 + 9900) = sqrt(16900) = 130 m. So the distance from H to CD being ~0.857 meters is impossible, since it's less than the distance from H to C or D. Therefore, there must be a fundamental error in my approach.

Wait, I think I forgot that the distance from H to line CD is not just the minimal HP, but in 3D space, the line CD is on the ground, and H is above the ground. So the minimal distance from H to line CD should be the length of the perpendicular from H to line CD, which would be the same as the height from H to the line in 3D. However, since the line is on the ground, this distance should be related to the height h and the horizontal distance from H's projection to the line CD. Let me think.

The projection of H onto the ground is O. So the horizontal distance from O to line CD is the distance from O to CD. Then, the minimal distance from H to CD would be the hypotenuse of a right triangle with legs equal to the height h and the horizontal distance from O to CD. Wait, is that true?

Let me recall that in 3D, the distance from a point to a line can be found by considering the distance from the projection of the point to the line, and then using Pythagoras. If the point is at height h, and the horizontal distance from the projection to the line is d_h, then the minimal distance is sqrt(d_h² + h²).

Therefore, maybe I can compute the horizontal distance from O to line CD, then the minimal distance from H to CD is sqrt(h² + d_h²).

Let me try this approach.

First, compute the horizontal distance from O to line CD. Since O is at (0, 0), and line CD goes from C(0, -c) to D(d, 0). The distance from O to line CD can be computed using the formula for distance from a point to a line in 2D.

The line CD can be represented as ax + by + c = 0. Let me find its equation.

Points C(0, -c) and D(d, 0). The slope of CD is (0 - (-c))/(d - 0) = c/d. So the equation of CD is y + c = (c/d)(x - 0) → y = (c/d)x - c.

Rewriting: (c/d)x - y - c = 0. Multiply both sides by d: cx - dy - cd = 0.

Therefore, the equation is cx - dy - cd = 0.

The distance from O(0, 0) to this line is |c*0 - d*0 - cd| / sqrt(c² + d²) = | -cd | / sqrt(c² + d²) = cd / sqrt(c² + d²).

Therefore, the horizontal distance from O to line CD is cd / sqrt(c² + d²).

Therefore, the minimal distance from H to line CD is sqrt( (cd / sqrt(c² + d²))² + h² ) = sqrt( c²d² / (c² + d²) + h² ).

Let me compute this:

= sqrt( [c²d² + h²(c² + d²)] / (c² + d²) )

Which is the same as sqrt( [h²(c² + d²) + c²d²] / (c² + d²) )

But this is different from what I computed earlier. Previously, I had divided by (c² + d²)^2, but now I have divided by (c² + d²).

Wait, so what's the correct formula?

If the horizontal distance from O to CD is cd / sqrt(c² + d²). Then, since H is directly above O at height h, the minimal distance from H to CD is the hypotenuse of a right triangle with legs h and cd / sqrt(c² + d²). Therefore, distance squared is h² + (cd / sqrt(c² + d²))² = h² + c²d² / (c² + d²).

Which is equal to [h²(c² + d²) + c²d²] / (c² + d²).

Therefore, distance is sqrt([h²(c² + d²) + c²d²] / (c² + d²)) = sqrt([h²(c² + d²) + c²d²]) / sqrt(c² + d²).

Which is different from what I calculated before. Earlier, I had divided by (c² + d²)^2, but that was a mistake.

Ah! So my previous calculation of distance squared as [h²(c² + d²) + c²d²]/(c² + d²)^2 was incorrect. It should be divided by (c² + d²), not squared.

So, correct expression is sqrt([h²(c² + d²) + c²d²]/(c² + d²)).

Let me compute this with our numbers:

Numerator inside sqrt: h²(c² + d²) + c²d² = 282,240,000.

Denominator: c² + d² = 19600.

So distance = sqrt(282,240,000 / 19600).

Compute 282,240,000 / 19600 = 282,240,000 / 19600 = let's divide numerator and denominator by 100: 2,822,400 / 196.

Divide numerator and denominator by 4: 705,600 / 49.

Now, 49 * 14,000 = 686,000. 705,600 - 686,000 = 19,600. 49 * 400 = 19,600. So total is 14,000 + 400 = 14,400.

Therefore, distance = sqrt(14,400) = 120 meters!

Ah! There we go. Finally, this makes sense. The minimal distance from H to line CD is 120 meters.

So that was the mistake. Earlier, I incorrectly divided by (c² + d²)^2 instead of (c² + d²).

So the minimal HP is 120 meters. Therefore, the original total length of HC and HD is 150 + 130 = 280 meters. If we replace them with a single rope HP of 120 meters, the length saved is 280 - 120 = 160 meters.

Wait, but the problem says to determine the greatest length of rope that can be saved. So is 160 meters the answer? But let me verify this result.

Let me check the calculation again:

We found that the minimal distance from H to CD is 120 meters, which is achieved at t = 9/14 on the segment CD. So this is valid, because earlier we found that t_min = c² / 19600 = 12600 / 19600 = 9/14, which is within [0, 1]. Therefore, the minimal HP is indeed 120 meters.

Therefore, the greatest length saved is 280 - 120 = 160 meters.

But wait, let me check with the earlier approach where we parametrized HP(t) = sqrt(t²d² + c²(1 - t)^2 + h²), and found that the minimum occurs at t = 9/14. Let's compute HP at t = 9/14.

Let me compute HP² at t = 9/14:

HP² = t²d² + c²(1 - t)^2 + h².

We can use the values:

t = 9/14, d² = 7000, c² = 12600, h² = 9900.

Compute t²d² = (81/196)*7000 = (81 * 7000)/196.

7000 / 196 = 7000 / 196 = 35.714... but let me calculate:

196 * 35 = 6860, 7000 - 6860 = 140, 140/196 = 0.714285... So total 35 + 140/196 = 35 + 5/7 = 35.714285...

But let me compute as fractions:

7000 / 196 = 7000 / 196 = 7000 / 196 = divide numerator and denominator by 28: 250 / 7.

So t²d² = (81/196) * 7000 = 81 * (7000 / 196) = 81 * (250 / 7) = (81 * 250) / 7 = 20250 / 7 ≈ 2892.857.

Similarly, c²(1 - t)^2 = 12600 * (1 - 9/14)^2 = 12600 * (5/14)^2 = 12600 * 25/196 = 12600 / 196 * 25.

12600 / 196 = 64.2857... Let me compute it as fractions: 12600 / 196 = divide numerator and denominator by 28: 450 / 7.

So c²(1 - t)^2 = (450 / 7) * 25 = 11250 / 7 ≈ 1607.142.

Then h² = 9900.

So total HP² = (20250 + 11250)/7 + 9900 = (31500 / 7) + 9900 = 4500 + 9900 = 14400.

Therefore, HP = sqrt(14400) = 120. Yes! This confirms that the minimal HP is indeed 120 meters.

Therefore, the length saved is 280 - 120 = 160 meters.

But let me check again if there are any mistakes.

Original ropes HC (150) and HD (130) total 280. Replaced by HP of 120, saving 160 meters. This seems correct.

But let me think if there's another way to get a shorter HP. For example, if the minimal HP is achieved at the foot of the perpendicular, which is inside CD, then that is indeed the minimal. We have confirmed with two methods that HP_min = 120. Hence, the answer should be 160 meters.

But let me check with another approach. Since we have from earlier:

The horizontal distance from O to CD is cd / sqrt(c² + d²).

With c = sqrt(12600), d = sqrt(7000), c² + d² = 19600, so sqrt(c² + d²) = 140.

So horizontal distance is cd / 140.

cd = sqrt(12600) * sqrt(7000) = sqrt(12600 * 7000). Let me compute 12600 * 7000.

But earlier we had c²d² = 12600 * 7000 = 88,200,000. Therefore, cd = sqrt(88,200,000) = sqrt(88,200,000). Let me calculate that:

88,200,000 = 88.2 * 10^6 = (9.4 * 10^0) * 10^6? Let me compute sqrt(88,200,000).

Note that sqrt(88,200,000) = sqrt(88.2 * 10^6) = sqrt(88.2) * 10^3.

sqrt(88.2) ≈ 9.4. Let me calculate it more precisely.

9.4² = 88.36, which is slightly more than 88.2. So 9.4² = 88.36, so sqrt(88.2) ≈ 9.4 - (88.36 - 88.2)/(2*9.4) ≈ 9.4 - 0.16/18.8 ≈ 9.4 - 0.0085 ≈ 9.3915.

Therefore, cd ≈ 9.3915 * 10^3 = 9391.5 meters.

Then horizontal distance from O to CD is cd / 140 ≈ 9391.5 / 140 ≈ 67.08 meters.

Then the minimal distance from H to CD is sqrt(h² + (horizontal distance)^2) = sqrt(9900 + (67.08)^2).

Calculate 67.08² ≈ 4500.

So sqrt(9900 + 4500) = sqrt(14400) = 120 meters. Correct!

Therefore, this confirms the result. Hence, the minimal HP is 120 meters, so the greatest length saved is 160 meters.

Therefore, the answer is 160 meters.

**Final Answer**
\boxed{160}

To solve this problem, we are given:

- A hot-air balloon at point $ H $, directly above the center point $ O $ of a flat field.
- Points $ A, B, C, D $ are positioned around $ O $, with:
  - $ A $: north of $ O $
  - $ B $: west of $ O $
  - $ C $: south of $ O $
  - $ D $: east of $ O $
  - $ C $ and $ D $ are 140 meters apart.
- The lengths of ropes $ HC $ and $ HD $ are 150 meters and 130 meters respectively.
- These two ropes are to be replaced by a single rope $ HP $, where $ P $ lies somewhere on the line segment $ CD $, to **minimize the total length of rope used**.
- We are asked to find the **maximum possible savings** in the length of rope.

---

### Step 1: Coordinate Setup

We place the points in a coordinate system:

- Let $ O $ = origin: $ (0, 0, 0) $
- Since $ H $ is directly above $ O $, it lies on the vertical axis: say $ H = (0, 0, h) $
- Then:
  - $ C = (0, -c, 0) $
  - $ D = (d, 0, 0) $
- Given that the distance between $ C $ and $ D $ is 140 m:
  $$
  \sqrt{d^2 + c^2} = 140 \Rightarrow d^2 + c^2 = 19600
  $$
- Also given:
  - $ HC = 150 \Rightarrow \sqrt{c^2 + h^2} = 150 \Rightarrow c^2 + h^2 = 22500 $
  - $ HD = 130 \Rightarrow \sqrt{d^2 + h^2} = 130 \Rightarrow d^2 + h^2 = 16900 $

From this, we can find:
- $ h^2 = 22500 - c^2 $
- $ h^2 = 16900 - d^2 $

So:
$$
22500 - c^2 = 16900 - d^2 \Rightarrow d^2 - c^2 = -5600
$$
But from earlier:
$$
d^2 + c^2 = 19600
$$
Solving these equations simultaneously:

Add and subtract:
- $ 2d^2 = 19600 - 5600 = 14000 \Rightarrow d^2 = 7000 $
- $ 2c^2 = 19600 + 5600 = 25200 \Rightarrow c^2 = 12600 $
- Then $ h^2 = 22500 - 12600 = 9900 \Rightarrow h = \sqrt{9900} = 99.499 $ meters

---

### Step 2: Optimal Location for $ P $ on Segment $ CD $

We want to minimize the length of rope $ HP $, where $ P $ is a point along segment $ CD $. The rope $ HC $ and $ HD $ together have length $ 150 + 130 = 280 $ meters. Our goal is to find the **minimum** possible length of $ HP $, and then compute the **maximum savings** as:

$$
\text{Savings} = 280 - \text{min}(HP)
$$

We use geometry to find the minimal distance from $ H $ to the line segment $ CD $.

---

### Step 3: Minimal Distance from $ H $ to Line $ CD $

Let’s compute the **shortest distance from point $ H $ to the line $ CD $**.

We can use the formula for the distance from a point to a line in 3D:

$$
\text{Distance} = \frac{|(\vec{u} \times \vec{v})|}{|\vec{v}|}
$$

Where:

- $ \vec{u} = \vec{HC} = (0, -c, -h) $
- $ \vec{v} = \vec{CD} = (d, c, 0) $

So the cross product $ \vec{u} \times \vec{v} $ is:

$$
\vec{u} \times \vec{v} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & -c & -h \\
d & c & 0
\end{vmatrix}
= (hc, -hd, cd)
$$
$$
|\vec{u} \times \vec{v}| = \sqrt{(hc)^2 + (hd)^2 + (cd)^2}
$$

$$
|\vec{v}| = \sqrt{d^2 + c^2} = \sqrt{19600} = 140
$$

So the distance from $ H $ to the line $ CD $ is:

$$
\text{Distance} = \frac{\sqrt{c^2h^2 + d^2h^2 + c^2d^2}}{140}
$$

Now plug in the known values:
- $ c^2 = 12600 $
- $ d^2 = 7000 $
- $ h^2 = 9900 $
- $ c^2d^2 = 12600 \cdot 7000 = 88{,}200{,}000 $
- $ c^2h^2 = 12600 \cdot 9900 = 124{,}740{,}000 $
- $ d^2h^2 = 7000 \cdot 9900 = 69{,}300{,}000 $

So:

$$
c^2h^2 + d^2h^2 + c^2d^2 = 124{,}740{,}000 + 69{,}300{,}000 + 88{,}200{,}000 = 282{,}240{,}000
$$

$$
\text{Distance} = \frac{\sqrt{282{,}240{,}000}}{140} = \frac{16{,}800}{140} = 120 \text{ meters}
$$

---

### Step 4: Compute the Savings

The total length of the two ropes $ HC $ and $ HD $ is:

$$
HC + HD = 150 + 130 = 280 \text{ meters}
$$

The optimal single rope $ HP $ can be as short as **120 meters**, so the **maximum savings** is:

$$
280 - 120 = \boxed{160}
$$

---

### Final Answer

$$
\boxed{160}
$$

[Assistant]
Explanation:
Okay, so there's this problem about a hot-air balloon with ropes attached to four points around a central point O. The balloon is directly above O, and there are points A, B, C, D arranged around O: A is north, B is west, C is south, D is east. The distance between C and D is 140 meters. The ropes HA, HB, HC, HD are all connected from the balloon H to these points. Now, they want to replace ropes HC and HD with a single rope HP, where P is on the straight line between C and D. The goal is to figure out the greatest length of rope that can be saved by doing this replacement.

First, let me visualize this. Since A is north of O, B west, C south, D east, so O is the center. So if I imagine a coordinate system with O at (0,0), then A would be (0, y), B (-x, 0), C (0, -z), D (w, 0). But maybe coordinates can help here. The Asymptote code given probably has coordinates, but since it's not directly helpful, maybe I need to assign coordinates to figure out the positions.

The distance between C and D is 140 meters. If C is directly south of O, then coordinates of C would be (0, -c), and D is directly east of O, so (d, 0). Then the distance between C and D would be sqrt((d - 0)^2 + (0 - (-c))^2) = sqrt(d² + c²) = 140 m. So that's one equation.

Also, rope HC is 150 m and rope HD is 130 m. Since H is directly above O, let's say H has coordinates (0, 0, h) if we consider O as (0,0,0). But since the problem is on a flat field, maybe we can just treat it as 2D, with H being at (0, 0, h), and the points A, B, C, D being on the ground. Then the lengths of the ropes HA, HB, HC, HD are the distances from H to each of those points.

So if we consider H at (0, 0, h), then HC would be the distance from (0, 0, h) to C (which is, say, (0, -c, 0) if C is directly south of O. Wait, but coordinates can be assigned in such a way. Let me try to formalize this.

Let me set up coordinates. Let me assume O is at (0, 0). Let me define coordinates for A, B, C, D:

- A is directly north of O: so coordinates (0, a) for some a.

- B is directly west of O: coordinates (-b, 0) for some b.

- C is directly south of O: (0, -c) for some c.

- D is directly east of O: (d, 0) for some d.

Now, the distance between C and D is given as 140 m. So the distance between C(0, -c) and D(d, 0) is sqrt[(d - 0)^2 + (0 - (-c))^2] = sqrt(d² + c²) = 140. So we have that equation.

Now, the ropes from H to the points. Since H is directly above O, H must be at (0, 0, h) for some height h. Then the length of HC is the distance from H(0,0,h) to C(0, -c, 0), which is sqrt[(0 - 0)^2 + (-c - 0)^2 + (0 - h)^2] = sqrt(c² + h²) = 150 m. Similarly, HD is the distance from H to D(d, 0, 0): sqrt[(d - 0)^2 + (0 - 0)^2 + (0 - h)^2] = sqrt(d² + h²) = 130 m.

So we have two equations:

1. sqrt(c² + h²) = 150 → c² + h² = 150² = 22500.

2. sqrt(d² + h²) = 130 → d² + h² = 130² = 16900.

Also, from the distance between C and D:

sqrt(d² + c²) = 140 → d² + c² = 140² = 19600.

Wait, hold on! Wait, if sqrt(d² + c²) = 140, then d² + c² = 19600. But we also have from equations 1 and 2:

From equation 1: c² + h² = 22500.

From equation 2: d² + h² = 16900.

If we subtract the second equation from the first, we get (c² + h²) - (d² + h²) = 22500 - 16900 → c² - d² = 5600.

But from the distance between C and D: d² + c² = 19600.

So now we have:

c² - d² = 5600,

c² + d² = 19600.

If we add these two equations: 2c² = 5600 + 19600 = 25200 → c² = 12600 → c = sqrt(12600) = sqrt(100 * 126) = 10*sqrt(126) = 10*sqrt(9*14) = 10*3*sqrt(14) = 30√14 ≈ ... but maybe exact value is not needed yet.

Similarly, subtracting the first equation from the second: (c² + d²) - (c² - d²) = 19600 - 5600 → 2d² = 14000 → d² = 7000 → d = sqrt(7000) = sqrt(100*70) = 10*sqrt(70).

So now we can find h² from equation 2: d² + h² = 16900 → h² = 16900 - d² = 16900 - 7000 = 9900 → h = sqrt(9900) = sqrt(100*99) = 10*sqrt(99) = 10*sqrt(9*11) = 30√11. Wait, sqrt(99) is 3√11? Wait, sqrt(99) is sqrt(9*11) which is 3√11. Yes. So h is 30√11 meters.

So now, we know c, d, h. But maybe we don't need the exact values? Wait, maybe we need coordinates for P?

But the problem is about replacing ropes HC and HD with a single rope HP where P is on CD. So the original total length of ropes HC and HD is 150 + 130 = 280 m. The new length would be HP, so the amount saved is 280 - HP. We need to find the maximum possible value of this, which is equivalent to minimizing HP. Therefore, the greatest length that can be saved is 280 - min(HP). Therefore, we need to find the minimal HP where P is on segment CD.

Therefore, the problem reduces to: find the minimal distance from H to the line segment CD, and then subtract that from 280.

But wait, hold on. However, since H is above O, and CD is a line segment between C and D. So HP is the straight line from H to P, where P is somewhere on CD. Therefore, the minimal HP is the minimal distance from H to the line CD, provided that the foot of the perpendicular from H to CD lies on the segment CD. If it does, then that's the minimal distance. Otherwise, the minimal distance would be the distance to the nearest endpoint. Therefore, first we need to check if the perpendicular from H to CD intersects CD. If so, then the minimal HP is that distance. If not, then it's the distance to C or D.

So let's find coordinates of H, C, D, then find the minimal distance from H to CD.

Earlier, we had coordinates for C(0, -c), D(d, 0), and H is at (0, 0, h). Wait, but in the coordinate system, if we are working in 2D, then H is at (0, 0) with height h? Wait, actually, since we are dealing with distances in 3D (since H is above the ground), but the points A, B, C, D are on the ground. So the coordinates for C and D are (0, -c, 0) and (d, 0, 0), and H is (0, 0, h). Then the line CD is from (0, -c, 0) to (d, 0, 0). The point P is somewhere on that line. The distance from H to P would be the length of HP, which is sqrt( (x_P - 0)^2 + (y_P - 0)^2 + (0 - h)^2 ) where (x_P, y_P, 0) is a point on CD.

Alternatively, since P lies on CD, we can parametrize CD. Let me parametrize CD. Let me consider CD as a line segment between C(0, -c, 0) and D(d, 0, 0). Let me parametrize P as follows: for t in [0, 1], P(t) = C + t*(D - C) = (0 + t*d, -c + t*(0 - (-c)), 0) = (td, -c + tc, 0) = (td, -c(1 - t), 0).

Therefore, coordinates of P(t) are (td, -c(1 - t), 0).

Then, the distance HP is sqrt( (td - 0)^2 + (-c(1 - t) - 0)^2 + (0 - h)^2 ) = sqrt( t²d² + c²(1 - t)² + h² ).

Therefore, HP² = t²d² + c²(1 - t)² + h². To minimize HP, we can minimize HP².

So let's consider f(t) = t²d² + c²(1 - t)^2 + h².

Let me compute f(t):

f(t) = d² t² + c² (1 - 2t + t²) + h² = (d² + c²) t² - 2c² t + c² + h².

But we already know that d² + c² = 19600. Therefore, f(t) = 19600 t² - 2c² t + c² + h².

So to find the minimum of this quadratic function in t. Since it's a quadratic in t, the minimum occurs at t = [2c²]/[2*19600] = c² / 19600.

Wait, more precisely, for a quadratic function f(t) = at² + bt + c, the vertex is at t = -b/(2a).

In our case, f(t) = 19600 t² - 2c² t + (c² + h²). Therefore, a = 19600, b = -2c². So the minimum occurs at t = (2c²)/(2*19600) = c² / 19600.

Therefore, t_min = c² / 19600.

But we need to check if t_min is within [0, 1], since t is between 0 and 1 for the segment CD.

So let's compute t_min:

We have c² = 12600 (from earlier: c² = 12600). Therefore, t_min = 12600 / 19600 = 126 / 196 = 9/14 ≈ 0.6428... which is between 0 and 1. Therefore, the foot of the perpendicular from H to CD lies on the segment CD. Therefore, the minimal HP is achieved at t = 9/14.

Therefore, the minimal HP is sqrt(f(t_min)).

Alternatively, since we can compute f(t_min):

f(t_min) = 19600*(c² / 19600)^2 - 2c²*(c² / 19600) + c² + h².

Wait, maybe there is a better way. Since we know that H is at (0, 0, h), and P lies on CD, then the minimal distance from H to CD is the length of the perpendicular from H to line CD. However, since we have coordinates, maybe we can compute that.

Alternatively, since we have already parametrized it, maybe we can compute f(t_min). Let me try another approach.

Alternatively, since we have coordinates of H, C, D, we can find the equation of line CD and then compute the distance from H to line CD.

But since H is not on the same plane as CD? Wait, CD is on the ground (z=0), and H is at (0, 0, h). So the distance from H to line CD is the minimal distance between H and any point on CD. Which is the same as sqrt( (distance from H to projection on plane)^2 + (height)^2 )? Wait, no, since the line CD is on the plane z=0, and H is at (0,0,h). The minimal distance from H to line CD is the minimal distance between H and points on CD. Which is the same as the minimal sqrt( (x)^2 + (y)^2 + h² ) where (x, y) is on line CD.

But since (x, y) is on CD, which goes from C(0, -c) to D(d, 0). So parametrizing CD as earlier: (td, -c(1 - t)).

Therefore, distance squared is (td)^2 + (-c(1 - t))^2 + h², same as before. So yes, so the minimal value is f(t_min) as above.

But maybe we can compute it more easily. Let me recall that in 3D, the distance from a point to a line can be calculated by the formula:

Distance = |(P0 - P) × v| / |v|,

where P is the point, P0 is a point on the line, and v is the direction vector of the line.

So let's try that. Let me take P as H(0, 0, h). Let me take P0 as point C(0, -c, 0). The direction vector of CD is D - C = (d, 0, 0) - (0, -c, 0) = (d, c, 0). So direction vector v = (d, c, 0).

Then vector P0 - P is C - H = (0 - 0, -c - 0, 0 - h) = (0, -c, -h).

Then the cross product (P0 - P) × v is determinant:

i   j   k

0  -c  -h

d   c   0

Calculating determinant:

i * [(-c)(0) - (-h)(c)] - j * [0*0 - (-h)*d] + k * [0*c - (-c)*d]

= i*(0 + hc) - j*(0 + hd) + k*(0 + cd)

= (hc, -hd, cd)

Then the magnitude of this cross product is sqrt( (hc)^2 + (-hd)^2 + (cd)^2 ) = sqrt( h²c² + h²d² + c²d² )

Then divide by |v|, which is sqrt(d² + c²) = 140.

So distance is sqrt( h²c² + h²d² + c²d² ) / 140.

Therefore, distance squared is [ h²c² + h²d² + c²d² ] / 140².

But wait, distance squared is (hc)^2 + (hd)^2 + (cd)^2 divided by 140²?

Wait, no. Wait, the distance is |(P0 - P) × v| / |v|. So squared distance is [ |(P0 - P) × v|² ] / |v|².

So |(P0 - P) × v|² is (hc)^2 + (hd)^2 + (cd)^2. Therefore, squared distance is [ h²c² + h²d² + c²d² ] / (d² + c²)².

But since d² + c² = 19600, so denominator is 19600².

But let me compute numerator:

h²c² + h²d² + c²d² = h²(c² + d²) + c²d².

From earlier, c² + d² = 19600, and c²d² is (cd)^2. Wait, but we can compute h²(c² + d²) + c²d². Let's compute:

We know c² + d² = 19600.

From equations 1 and 2:

From equation 1: c² + h² = 22500 → h² = 22500 - c².

Similarly, equation 2: d² + h² = 16900 → h² = 16900 - d².

But maybe another way. Let's compute h²(c² + d²) + c²d²:

= h²*(c² + d²) + c²d².

We have h² = ?

Earlier, we found h² = 9900. Wait, from equation 2: d² + h² = 16900. Since d² = 7000, then h² = 16900 - 7000 = 9900. Yes. So h² = 9900. Therefore:

h²*(c² + d²) + c²d² = 9900 * 19600 + c²d².

But we also can compute c²d². Since c² = 12600, d² = 7000, so c²d² = 12600 * 7000.

Therefore:

Numerator is 9900 * 19600 + 12600 * 7000.

Let me compute this. Let me note that 12600 = 126 * 100, 7000 = 70 * 100, 9900 = 99 * 100, 19600 = 196 * 100. So maybe factor out 100^2:

Numerator = (99 * 196 * 100^2) + (126 * 70 * 100^2) = 100^2 [99*196 + 126*70].

Compute 99*196:

Well, 100*196 = 19600, minus 1*196 = 19600 - 196 = 19404.

Then 126*70 = (120 + 6)*70 = 120*70 + 6*70 = 8400 + 420 = 8820.

So total inside the brackets is 19404 + 8820 = 28224.

Therefore, numerator is 100^2 * 28224. Therefore, squared distance is [100² * 28224] / [19600²] = [28224 / 19600²] * 100² = ?

Wait, let me calculate:

Squared distance = [Numerator] / [denominator] = [100² * 28224] / [19600²] = (28224) / (19600² / 100²) ) = 28224 / ( (19600 / 100)^2 ) = 28224 / (196²).

Compute 196²: 196*196. Well, 200² = 40000, minus 4*200 + 4 = 40000 - 800 + 4 = 39204? Wait, no. Wait, (a - b)^2 = a² - 2ab + b². Let me compute 196²:

196 * 196:

Well, 200 * 196 = 39200,

Minus 4 * 196 = 784,

So 39200 - 784 = 38416. Yes, 196² = 38416.

Therefore, squared distance = 28224 / 38416.

Simplify this fraction. Let's divide numerator and denominator by 16:

28224 / 16 = 1764,

38416 / 16 = 2401.

So 1764 / 2401.

Check if they can be simplified further. Let me check GCD of 1764 and 2401.

Factor 1764: 42² = (6*7)^2 = 6²*7² = 36*49. Also, 1764 is 49*36 = 49*36.

Factor 2401: 49*49 = 2401. So 2401 = 49². Therefore, 1764 = 49*36, 2401 = 49². Therefore, simplify 1764 / 2401 = (49*36)/(49*49) ) = 36 / 49.

Therefore, squared distance is 36/49. Therefore, distance is sqrt(36/49) = 6/7. Wait, but what are the units here? Because the numerator and denominator?

Wait, hold on. Wait, we computed squared distance as [Numerator]/[Denominator] = 28224 / 38416 = 36/49. So sqrt(36/49) = 6/7. But what are the units?

Wait, actually, we had:

Squared distance = [ |(P0 - P) × v|² ] / |v|² = [ h²c² + h²d² + c²d² ] / (d² + c²)^2 = 36/49.

But wait, that would mean the distance is 6/7? But that seems odd because all the lengths were in meters. Wait, no, because we had earlier calculated numerator as (h²c² + h²d² + c²d²) which is in units of meters squared. Wait, no, let me retrace.

Wait, when we computed the numerator, which was |(P0 - P) × v|² = h²c² + h²d² + c²d². But we computed that as 100² * 28224, but in reality, those numbers were in terms of the variables.

But actually, perhaps I made a miscalculation here. Because when I calculated:

Numerator = h²(c² + d²) + c²d².

But h² = 9900, c² + d² = 19600, and c²d² = 12600 * 7000.

So h²(c² + d²) = 9900 * 19600, and c²d² = 12600 * 7000.

But when we did the calculation, we considered the numerical values. Wait, but in reality, that expression is:

Numerator is h²(c² + d²) + c²d².

But we can compute this as:

= h²*(c² + d²) + c²d².

But since h² = 22500 - c², from equation 1:

Wait, equation 1 was c² + h² = 22500, so h² = 22500 - c². Therefore, substitute:

Numerator = (22500 - c²)(c² + d²) + c²d².

But c² + d² = 19600, so:

= (22500 - c²)(19600) + c²d².

But we also know that d² = 19600 - c². So:

= (22500 - c²)(19600) + c²(19600 - c²)

= 22500*19600 - c²*19600 + 19600 c² - c^4

= 22500*19600 - c^4.

So the numerator is 22500*19600 - c^4.

But c² = 12600, so c^4 = (12600)^2.

So numerator = 22500*19600 - (12600)^2.

Hmm, this seems more complicated. However, earlier calculation said that this equals 100² * 28224. Maybe there was a miscalculation.

Wait, but if we use the previous approach where we found that squared distance is 36/49, that would mean that the distance from H to line CD is 6/7 of some unit? But this seems contradictory.

Alternatively, maybe I messed up the cross product calculation? Let me check again:

Earlier, cross product (P0 - P) × v was (hc, -hd, cd). Then its magnitude squared is h²c² + h²d² + c²d². Then divided by |v|² which is (d² + c²). Therefore, squared distance is [h²(c² + d²) + c²d²]/(c² + d²)^2.

But that's equal to [h² + (c²d²)/(c² + d²)] / (c² + d²).

Wait, maybe not. Wait, more accurately, squared distance = [h²(c² + d²) + c²d²]/(c² + d²)^2 = [h²/(c² + d²)] + [c²d²/(c² + d²)^2].

But maybe we can compute this with our known values:

We have c² + d² = 19600, h² = 9900, c²d² = 12600 * 7000.

So let's compute h²(c² + d²) + c²d² = 9900 * 19600 + 12600 * 7000.

Let me compute 9900 * 19600:

Note that 9900 = 10000 - 100, so (10000 - 100)*19600 = 10000*19600 - 100*19600 = 196,000,000 - 1,960,000 = 194,040,000.

Then 12600 * 7000. Let's compute 12600 * 7000. 12600 * 7000 = 12.6 * 10^3 * 7 * 10^3 = 88.2 * 10^6 = 88,200,000.

Therefore, total numerator is 194,040,000 + 88,200,000 = 282,240,000.

Denominator is (c² + d²)^2 = (19600)^2 = 384,160,000.

Therefore, squared distance is 282,240,000 / 384,160,000 = 282,240 / 384,160 = divide numerator and denominator by 160: 282,240 ÷ 160 = 1764, 384,160 ÷ 160 = 2401. So 1764 / 2401 as before. Which reduces to 36 / 49. Therefore, squared distance is 36/49, so distance is 6/7 meters? Wait, no. Wait, the units here are?

Wait, the numerator was h²c² + h²d² + c²d², which has units of (m² * m²) = m⁴? Wait, no. Wait, h is in meters, c is in meters, so h²c² is (m²)(m²) = m⁴. Similarly, h²d² is m⁴, c²d² is m⁴. Then the numerator is m⁴, denominator is (m²)^2 = m⁴. Therefore, distance squared is dimensionless? Wait, no:

Wait, the distance is in meters. Wait, no. Wait, the cross product [(P0 - P) × v] has units of meters^3? Wait, original cross product: (P0 - P) is a vector from H to C, which is in meters in each component. v is direction vector of CD, also in meters. Therefore, cross product has units of meters^2. Then |(P0 - P) × v| has units meters². Divided by |v| (meters), so the distance has units meters. Therefore, squared distance has units meters². Therefore, our earlier calculation of 36/49 must be in units of meters squared? But how?

Wait, we did the computation with actual numbers:

Numerator was 282,240,000 m⁴? Wait, no. Wait, the actual calculation:

Wait, let me think again. Let me check with specific numbers. Let me think of all variables in meters:

h is in meters, c and d are in meters. Therefore, h²c² has units of meters^4. Therefore, the numerator [h²c² + h²d² + c²d²] has units of meters^4. The denominator (c² + d²)^2 is meters^4. Therefore, the ratio is dimensionless? But then how come we get a numerical value of 36/49?

Wait, there must be a miscalculation here. Because if the distance squared is 36/49, then that would mean the distance is 6/7 meters, but that can't be, since HC is 150 meters, which is much longer. Therefore, clearly, I messed up the calculation of the cross product.

Wait, where did the 36/49 come from? It seems the problem is that I forgot that the cross product is calculated in 3D space. Wait, in our case, the cross product vector was (hc, -hd, cd). Then its magnitude squared is (hc)^2 + (-hd)^2 + (cd)^2 = h²c² + h²d² + c²d². But then divided by |v|² which is (d² + c²). Therefore, the squared distance is [h²(c² + d²) + c²d²]/(c² + d²)^2.

But this is equal to [h²/(c² + d²)] + [c²d²/(c² + d²)^2].

So let's compute that with numbers:

h² = 9900, c² + d² = 19600, c²d² = 12600 * 7000 = 88,200,000.

So:

First term: h²/(c² + d²) = 9900 / 19600 ≈ 0.5051.

Second term: c²d²/(c² + d²)^2 = 88,200,000 / (19600)^2. Let me compute 19600² = 384,160,000. So 88,200,000 / 384,160,000 ≈ 0.2297.

Total ≈ 0.5051 + 0.2297 ≈ 0.7348. Which is approximately 0.7348, which is roughly 0.7348. What is 36/49? 36 divided by 49 is approximately 0.73469... Yes! So 0.7348 ≈ 36/49. So that's correct. Therefore, squared distance is 36/49, so distance is 6/7 meters? Wait, no! Wait, no! Wait, squared distance is 36/49, but that's in meters squared? No, no! Wait, the actual calculation is:

The distance squared is [h²(c² + d²) + c²d²]/(c² + d²)^2. But this is equal to (from previous) 36/49. However, this is a dimensionless quantity? No! Wait, no, the numerator and denominator both have units of meters⁴? No, no, actually, c² + d² is meters², so (c² + d²)^2 is meters⁴, and the numerator is h²(c² + d²) + c²d², which is (m² * meters²) + meters⁴ = meters⁴ + meters⁴ = meters⁴. Therefore, the ratio is (meters⁴) / (meters⁴) = dimensionless. But we have a numerical value of 36/49. Which means that the squared distance is 36/49*(some unit)? No, no, this is not correct. Wait, there must be an error in the calculation. How can the distance squared be dimensionless?

Wait, no! Wait, no, the formula is:

Distance squared = [ |(P0 - P) × v|² ] / |v|².

Which is [ (meters²)² ] / (meters²) ) = meters² / meters² = dimensionless? No! Wait, no. Wait, (P0 - P) is a vector in 3D space, with units of meters. v is also a vector with units of meters. The cross product (P0 - P) × v has units of meters × meters = meters². Then squared magnitude is meters⁴. Divided by |v|² which is meters². Therefore, the distance squared has units of meters². Therefore, the numerical value 36/49 is multiplied by (meters²)? Wait, no. Wait, if I compute [h²(c² + d²) + c²d²]/(c² + d²)^2, which is 36/49, then that is equal to distance squared? But how can that be?

Ah! Wait, no, there was a mistake in the previous cross product calculation. Wait, the cross product [(P0 - P) × v] has magnitude |(P0 - P) × v|, which is equal to |P0 - P| * |v| * sin(theta), where theta is the angle between them. Therefore, the squared magnitude is |P0 - P|² * |v|² * sin²(theta). Then, divided by |v|² gives |P0 - P|² * sin²(theta). Which is equal to the squared distance from H to line CD.

Wait, but in reality, the distance from H to line CD is equal to |(P0 - P) × v| / |v|. Therefore, the squared distance is |(P0 - P) × v|² / |v|².

But according to our calculation, this equals 36/49. However, this suggests that the squared distance is 36/49 in meters squared? But 36/49 is just a number. Therefore, we must have forgotten to multiply by something?

Wait, no. Wait, the calculation of [h²(c² + d²) + c²d²]/(c² + d²)^2 resulted in 36/49. But that 36/49 is actually [(h²(c² + d²) + c²d²)] / (c² + d²)^2. However, in reality, this must have units. But since we plugged in the actual numerical values, the result is a numerical value times meters squared? No, no. Wait, let me think again.

Wait, maybe there was a miscalculation earlier where I thought the numerator was 282,240,000 and denominator was 384,160,000. Wait, but these numbers were in terms of the actual variables. Wait, actually, if we compute:

Numerator: h²(c² + d²) + c²d² = 9900 * 19600 + 12600 * 7000.

We calculated that as 194,040,000 + 88,200,000 = 282,240,000.

Denominator: (c² + d²)^2 = 19600² = 384,160,000.

Therefore, distance squared is 282,240,000 / 384,160,000 = 0.73469... which is 36/49. Therefore, distance is sqrt(36/49) = 6/7. But what is the unit? Since all the terms in numerator and denominator have units of meters⁴? Wait, no, no. Wait, h² is meters², (c² + d²) is meters², so h²*(c² + d²) is meters⁴. Similarly, c²d² is meters⁴. Therefore, the numerator is meters⁴, denominator is (meters²)^2 = meters⁴. Therefore, the ratio is dimensionless, so distance squared is 36/49, meaning distance is 6/7 meters? That's not possible because the distance from H to line CD can't be less than the distance from H to C or H to D, which are 150 and 130 meters.

Wait, this must mean there is an error in my calculations. Because 6/7 meters is about 0.857 meters, which is way smaller than 130 meters. Therefore, there's a miscalculation.

Wait, but where? Let me check the cross product again.

The cross product (P0 - P) × v. P0 is C(0, -c, 0). P is H(0, 0, h). Therefore, vector P0 - P is C - H = (0 - 0, -c - 0, 0 - h) = (0, -c, -h).

Vector v is direction vector of CD: D - C = (d, c, 0).

So cross product is determinant:

i   j   k

0  -c  -h

d   c   0

Which is i*( (-c)(0) - (-h)(c) ) - j*( (0)(0) - (-h)(d) ) + k*( (0)(c) - (-c)(d) )

= i*(0 + hc) - j*(0 + hd) + k*(0 + cd)

= (hc, -hd, cd)

Yes, that's correct. Then the magnitude squared is h²c² + h²d² + c²d². Then divided by |v|² = d² + c². Therefore, the squared distance is [h²(c² + d²) + c²d²] / (c² + d²)^2.

But with actual numbers, we said that [h²(c² + d²) + c²d²] = 282,240,000 and (c² + d²)^2 = 384,160,000, giving 0.73469... which is 36/49.

But if this is supposed to be the squared distance, then distance is sqrt(36/49) = 6/7 meters. Which is impossible. Therefore, there must be a miscalculation in the numerical values.

Wait, let me check again the value of h²(c² + d²) + c²d²:

h² = 9900, c² + d² = 19600, c²d² = 12600*7000 = 88,200,000.

So h²*(c² + d²) = 9900 * 19600 = Let me compute this carefully.

First compute 9900 * 19600.

Note that 9900 * 19600 = (10000 - 100) * 19600 = 10000*19600 - 100*19600 = 196,000,000 - 1,960,000 = 194,040,000. Yes.

Then c²d² = 12600 * 7000. Let me compute that as 12600 * 7000.

Note that 12600 * 7000 = 12600 * 7 * 1000 = 88200 * 1000 = 88,200,000. Correct.

So total numerator is 194,040,000 + 88,200,000 = 282,240,000.

Denominator is (19600)^2 = 384,160,000.

So the ratio is 282,240,000 / 384,160,000 = dividing numerator and denominator by 1000: 282,240 / 384,160.

Divide numerator and denominator by 160: 282,240 ÷ 160 = 1764; 384,160 ÷ 160 = 2401.

So 1764 / 2401.

Now, 1764 is 42², and 2401 is 49². So 1764 / 2401 = (42/49)² = (6/7)² = 36/49.

Ah! Therefore, the squared distance is (6/7)². Wait, no, the squared distance is [h²(c² + d²) + c²d²]/(c² + d²)^2 = (6/7)^2. Wait, but this is the squared distance? Wait, no! Wait, no, the squared distance is [ |(P0 - P) × v|² ] / |v|² = (282,240,000) / (384,160,000) = 36/49. Therefore, distance is sqrt(36/49) = 6/7. But what are the units?

Wait, no! This is where the error is. The cross product [(P0 - P) × v] has magnitude sqrt(h²c² + h²d² + c²d²), which we calculated as sqrt(282,240,000). Wait, no. Wait, no. Wait, we had |(P0 - P) × v|² = 282,240,000. Then dividing by |v|² = (19600)^2 = 384,160,000 gives us (distance)^2 = 282,240,000 / 384,160,000 = 36/49. So distance = 6/7. But this is in what units?

Wait, the problem is that all the terms in the numerator and denominator are meters⁴, so dividing them gives a dimensionless quantity. Therefore, the numerical value 6/7 is in (m²)? No, no. Wait, actually, no. Wait, the formula is:

distance = |(P0 - P) × v| / |v|.

But |(P0 - P) × v| is sqrt(h²c² + h²d² + c²d²). Wait, but h, c, d are in meters. So h²c² has units of m⁴? No! Wait, h is in meters, c is in meters, so h²c² is meters² * meters² = meters⁴. Therefore, |(P0 - P) × v| has units of sqrt(m⁴) = meters². Then divide by |v| which is in meters, so overall distance has units of meters² / meters = meters. Therefore, the numerical value 6/7 is multiplied by meters?

Wait, I think I need to realize that in the calculation, we used specific values for h, c, d, but forgot to carry the units. Let me re-express this.

Let me assume that all lengths are in meters. Then h, c, d are in meters. Let me compute the numerator |(P0 - P) × v|² = h²c² + h²d² + c²d². This is in (m²)(m²) + (m²)(m²) + (m²)(m²) = m⁴.

Denominator |v|² is meters².

Therefore, the entire expression is [m⁴] / [m²] = m². Therefore, the distance squared is in m², so distance is in meters.

But how does the numerical value of 36/49 translate to actual meters? Because we calculated it as a ratio of numbers. So the numerical value is 36/49, but multiplied by what? Wait, we have to realize that the actual calculation of |(P0 - P) × v|² is equal to 282,240,000, which is meters⁴. Therefore, divided by |v|² = 384,160,000 meters², gives distance squared = 282,240,000 / 384,160,000 meters² = 0.73469... meters². Therefore, distance is sqrt(0.73469...) meters ≈ 0.857 meters. Wait, but why is this number?

Wait, but this contradicts the fact that HC is 150 meters. How can the distance from H to CD be only 0.857 meters? That doesn't make sense. So clearly, there is a miscalculation.

Wait, this suggests that I messed up the cross product. Maybe I confused the formula? Let me think again.

Wait, the formula for the distance from a point to a line in 3D is |(P0 - P) × v| / |v|, where P is the point, P0 is a point on the line, and v is the direction vector of the line.

But in our case, P is H(0, 0, h), P0 is C(0, -c, 0), v is D - C = (d, c, 0).

So the vector P0 - P is (0 - 0, -c - 0, 0 - h) = (0, -c, -h).

Then the cross product is:

(P0 - P) × v = |i   j   k|
               0  -c  -h
               d   c   0

= i * [(-c)(0) - (-h)(c)] - j * [0*0 - (-h)(d)] + k * [0*c - (-c)(d)]

= i * [0 + hc] - j * [0 + hd] + k * [0 + cd]

= (hc, -hd, cd)

The magnitude squared of this vector is (hc)^2 + (hd)^2 + (cd)^2 = h²c² + h²d² + c²d².

But this is supposed to be |(P0 - P) × v|².

Therefore, the distance from H to line CD is sqrt(h²c² + h²d² + c²d²) / sqrt(d² + c²).

Let me compute this expression with our numerical values.

First compute numerator: sqrt(h²c² + h²d² + c²d²).

We had h²c² + h²d² + c²d² = 282,240,000. Wait, no! Wait, h²c² + h²d² + c²d² is equal to 9900*12600 + 9900*7000 + 12600*7000? No, no! Wait, earlier we said h²c² + h²d² + c²d² = h²(c² + d²) + c²d² = 9900*19600 + 12600*7000 = 282,240,000. So that is in meters⁴. Therefore, sqrt(282,240,000) is in meters², and then divided by sqrt(19600²) = 19600 meters. Therefore, the distance is sqrt(282,240,000)/19600 meters.

Let me compute sqrt(282,240,000). Let's note that 282,240,000 = 282,240 * 1000. Wait, but sqrt(282,240,000) = sqrt(282,240 * 1000) = sqrt(282,240) * sqrt(1000). Wait, but this seems messy. Wait, but earlier we said that this is equal to 6/7 * something?

Wait, no. Let me instead recall that we had [h²(c² + d²) + c²d²]/(c² + d²)^2 = 36/49. Therefore, distance squared is 36/49. Wait, but that can't be. Wait, no! No, the distance squared is [h²(c² + d²) + c²d²]/(c² + d²)^2? No, the distance squared is [h²(c² + d²) + c²d²]/(c² + d²)^2? No, the distance squared is [ |(P0 - P) × v|² ] / |v|² = [h²(c² + d²) + c²d²]/(c² + d²)^2. Wait, but this is equal to 36/49? Which would mean that distance squared is 36/49, but I think the units are missing. Wait, no. If all the terms are in meters, then the [h²(c² + d²) + c²d²] is in meters⁴, and divided by (meters²)^2, which is meters⁴, giving a dimensionless quantity. But then, how can this be?

Wait, I think the error is that I assumed that the cross product's magnitude squared was equal to h²c² + h²d² + c²d², but in reality, no. Let me recast the calculation with actual numerical values. Let me take specific numbers for h, c, d.

Let me recall:

c² = 12600, d² = 7000, h² = 9900.

Therefore, c = sqrt(12600) ≈ 112.25 m, d = sqrt(7000) ≈ 83.666 m, h = sqrt(9900) ≈ 99.499 m.

So let's compute the numerator, h²c² + h²d² + c²d².

h²c² = 9900 * 12600 = let's compute 9900 * 12600.

Well, 10,000 * 12,600 = 126,000,000

Minus 100 * 12,600 = 1,260,000

So 126,000,000 - 1,260,000 = 124,740,000.

Similarly, h²d² = 9900 * 7000 = 69,300,000.

c²d² = 12,600 * 7,000 = 88,200,000.

So total numerator: 124,740,000 + 69,300,000 + 88,200,000 = 124,740,000 + 69,300,000 = 194,040,000 + 88,200,000 = 282,240,000.

Denominator: (c² + d²)^2 = (19600)^2 = 384,160,000.

So distance squared = 282,240,000 / 384,160,000 = 0.73469...

Now, distance = sqrt(0.73469) ≈ 0.857 meters. But how can this be? If H is at 99.5 meters above O, and C is 112.25 meters south of O, then the distance from H to C is sqrt( (0 - 0)^2 + (-c - 0)^2 + (0 - h)^2 ) = sqrt(c² + h²) = sqrt(12600 + 9900) = sqrt(22500) = 150 m. Similarly, distance from H to D is sqrt(d² + h²) = sqrt(7000 + 9900) = sqrt(16900) = 130 m. So the distance from H to CD being ~0.857 meters is impossible, since it's less than the distance from H to C or D. Therefore, there must be a fundamental error in my approach.

Wait, I think I forgot that the distance from H to line CD is not just the minimal HP, but in 3D space, the line CD is on the ground, and H is above the ground. So the minimal distance from H to line CD should be the length of the perpendicular from H to line CD, which would be the same as the height from H to the line in 3D. However, since the line is on the ground, this distance should be related to the height h and the horizontal distance from H's projection to the line CD. Let me think.

The projection of H onto the ground is O. So the horizontal distance from O to line CD is the distance from O to CD. Then, the minimal distance from H to CD would be the hypotenuse of a right triangle with legs equal to the height h and the horizontal distance from O to CD. Wait, is that true?

Let me recall that in 3D, the distance from a point to a line can be found by considering the distance from the projection of the point to the line, and then using Pythagoras. If the point is at height h, and the horizontal distance from the projection to the line is d_h, then the minimal distance is sqrt(d_h² + h²).

Therefore, maybe I can compute the horizontal distance from O to line CD, then the minimal distance from H to CD is sqrt(h² + d_h²).

Let me try this approach.

First, compute the horizontal distance from O to line CD. Since O is at (0, 0), and line CD goes from C(0, -c) to D(d, 0). The distance from O to line CD can be computed using the formula for distance from a point to a line in 2D.

The line CD can be represented as ax + by + c = 0. Let me find its equation.

Points C(0, -c) and D(d, 0). The slope of CD is (0 - (-c))/(d - 0) = c/d. So the equation of CD is y + c = (c/d)(x - 0) → y = (c/d)x - c.

Rewriting: (c/d)x - y - c = 0. Multiply both sides by d: cx - dy - cd = 0.

Therefore, the equation is cx - dy - cd = 0.

The distance from O(0, 0) to this line is |c*0 - d*0 - cd| / sqrt(c² + d²) = | -cd | / sqrt(c² + d²) = cd / sqrt(c² + d²).

Therefore, the horizontal distance from O to line CD is cd / sqrt(c² + d²).

Therefore, the minimal distance from H to line CD is sqrt( (cd / sqrt(c² + d²))² + h² ) = sqrt( c²d² / (c² + d²) + h² ).

Let me compute this:

= sqrt( [c²d² + h²(c² + d²)] / (c² + d²) )

Which is the same as sqrt( [h²(c² + d²) + c²d²] / (c² + d²) )

But this is different from what I computed earlier. Previously, I had divided by (c² + d²)^2, but now I have divided by (c² + d²).

Wait, so what's the correct formula?

If the horizontal distance from O to CD is cd / sqrt(c² + d²). Then, since H is directly above O at height h, the minimal distance from H to CD is the hypotenuse of a right triangle with legs h and cd / sqrt(c² + d²). Therefore, distance squared is h² + (cd / sqrt(c² + d²))² = h² + c²d² / (c² + d²).

Which is equal to [h²(c² + d²) + c²d²] / (c² + d²).

Therefore, distance is sqrt([h²(c² + d²) + c²d²] / (c² + d²)) = sqrt([h²(c² + d²) + c²d²]) / sqrt(c² + d²).

Which is different from what I calculated before. Earlier, I had divided by (c² + d²)^2, but that was a mistake.

Ah! So my previous calculation of distance squared as [h²(c² + d²) + c²d²]/(c² + d²)^2 was incorrect. It should be divided by (c² + d²), not squared.

So, correct expression is sqrt([h²(c² + d²) + c²d²]/(c² + d²)).

Let me compute this with our numbers:

Numerator inside sqrt: h²(c² + d²) + c²d² = 282,240,000.

Denominator: c² + d² = 19600.

So distance = sqrt(282,240,000 / 19600).

Compute 282,240,000 / 19600 = 282,240,000 / 19600 = let's divide numerator and denominator by 100: 2,822,400 / 196.

Divide numerator and denominator by 4: 705,600 / 49.

Now, 49 * 14,000 = 686,000. 705,600 - 686,000 = 19,600. 49 * 400 = 19,600. So total is 14,000 + 400 = 14,400.

Therefore, distance = sqrt(14,400) = 120 meters!

Ah! There we go. Finally, this makes sense. The minimal distance from H to line CD is 120 meters.

So that was the mistake. Earlier, I incorrectly divided by (c² + d²)^2 instead of (c² + d²).

So the minimal HP is 120 meters. Therefore, the original total length of HC and HD is 150 + 130 = 280 meters. If we replace them with a single rope HP of 120 meters, the length saved is 280 - 120 = 160 meters.

Wait, but the problem says to determine the greatest length of rope that can be saved. So is 160 meters the answer? But let me verify this result.

Let me check the calculation again:

We found that the minimal distance from H to CD is 120 meters, which is achieved at t = 9/14 on the segment CD. So this is valid, because earlier we found that t_min = c² / 19600 = 12600 / 19600 = 9/14, which is within [0, 1]. Therefore, the minimal HP is indeed 120 meters.

Therefore, the greatest length saved is 280 - 120 = 160 meters.

But wait, let me check with the earlier approach where we parametrized HP(t) = sqrt(t²d² + c²(1 - t)^2 + h²), and found that the minimum occurs at t = 9/14. Let's compute HP at t = 9/14.

Let me compute HP² at t = 9/14:

HP² = t²d² + c²(1 - t)^2 + h².

We can use the values:

t = 9/14, d² = 7000, c² = 12600, h² = 9900.

Compute t²d² = (81/196)*7000 = (81 * 7000)/196.

7000 / 196 = 7000 / 196 = 35.714... but let me calculate:

196 * 35 = 6860, 7000 - 6860 = 140, 140/196 = 0.714285... So total 35 + 140/196 = 35 + 5/7 = 35.714285...

But let me compute as fractions:

7000 / 196 = 7000 / 196 = 7000 / 196 = divide numerator and denominator by 28: 250 / 7.

So t²d² = (81/196) * 7000 = 81 * (7000 / 196) = 81 * (250 / 7) = (81 * 250) / 7 = 20250 / 7 ≈ 2892.857.

Similarly, c²(1 - t)^2 = 12600 * (1 - 9/14)^2 = 12600 * (5/14)^2 = 12600 * 25/196 = 12600 / 196 * 25.

12600 / 196 = 64.2857... Let me compute it as fractions: 12600 / 196 = divide numerator and denominator by 28: 450 / 7.

So c²(1 - t)^2 = (450 / 7) * 25 = 11250 / 7 ≈ 1607.142.

Then h² = 9900.

So total HP² = (20250 + 11250)/7 + 9900 = (31500 / 7) + 9900 = 4500 + 9900 = 14400.

Therefore, HP = sqrt(14400) = 120. Yes! This confirms that the minimal HP is indeed 120 meters.

Therefore, the length saved is 280 - 120 = 160 meters.

But let me check again if there are any mistakes.

Original ropes HC (150) and HD (130) total 280. Replaced by HP of 120, saving 160 meters. This seems correct.

But let me think if there's another way to get a shorter HP. For example, if the minimal HP is achieved at the foot of the perpendicular, which is inside CD, then that is indeed the minimal. We have confirmed with two methods that HP_min = 120. Hence, the answer should be 160 meters.

But let me check with another approach. Since we have from earlier:

The horizontal distance from O to CD is cd / sqrt(c² + d²).

With c = sqrt(12600), d = sqrt(7000), c² + d² = 19600, so sqrt(c² + d²) = 140.

So horizontal distance is cd / 140.

cd = sqrt(12600) * sqrt(7000) = sqrt(12600 * 7000). Let me compute 12600 * 7000.

But earlier we had c²d² = 12600 * 7000 = 88,200,000. Therefore, cd = sqrt(88,200,000) = sqrt(88,200,000). Let me calculate that:

88,200,000 = 88.2 * 10^6 = (9.4 * 10^0) * 10^6? Let me compute sqrt(88,200,000).

Note that sqrt(88,200,000) = sqrt(88.2 * 10^6) = sqrt(88.2) * 10^3.

sqrt(88.2) ≈ 9.4. Let me calculate it more precisely.

9.4² = 88.36, which is slightly more than 88.2. So 9.4² = 88.36, so sqrt(88.2) ≈ 9.4 - (88.36 - 88.2)/(2*9.4) ≈ 9.4 - 0.16/18.8 ≈ 9.4 - 0.0085 ≈ 9.3915.

Therefore, cd ≈ 9.3915 * 10^3 = 9391.5 meters.

Then horizontal distance from O to CD is cd / 140 ≈ 9391.5 / 140 ≈ 67.08 meters.

Then the minimal distance from H to CD is sqrt(h² + (horizontal distance)^2) = sqrt(9900 + (67.08)^2).

Calculate 67.08² ≈ 4500.

So sqrt(9900 + 4500) = sqrt(14400) = 120 meters. Correct!

Therefore, this confirms the result. Hence, the minimal HP is 120 meters, so the greatest length saved is 160 meters.

Therefore, the answer is 160 meters.

**Final Answer**
\boxed{160}

To solve this problem, we are given:

- A hot-air balloon at point $ H $, directly above the center point $ O $ of a flat field.
- Points $ A, B, C, D $ are positioned around $ O $, with:
  - $ A $: north of $ O $
  - $ B $: west of $ O $
  - $ C $: south of $ O $
  - $ D $: east of $ O $
  - $ C $ and $ D $ are 140 meters apart.
- The lengths of ropes $ HC $ and $ HD $ are 150 meters and 130 meters respectively.
- These two ropes are to be replaced by a single rope $ HP $, where $ P $ lies somewhere on the line segment $ CD $, to **minimize the total length of rope used**.
- We are asked to find the **maximum possible savings** in the length of rope.

---

### Step 1: Coordinate Setup

We place the points in a coordinate system:

- Let $ O $ = origin: $ (0, 0, 0) $
- Since $ H $ is directly above $ O $, it lies on the vertical axis: say $ H = (0, 0, h) $
- Then:
  - $ C = (0, -c, 0) $
  - $ D = (d, 0, 0) $
- Given that the distance between $ C $ and $ D $ is 140 m:
  $$
  \sqrt{d^2 + c^2} = 140 \Rightarrow d^2 + c^2 = 19600
  $$
- Also given:
  - $ HC = 150 \Rightarrow \sqrt{c^2 + h^2} = 150 \Rightarrow c^2 + h^2 = 22500 $
  - $ HD = 130 \Rightarrow \sqrt{d^2 + h^2} = 130 \Rightarrow d^2 + h^2 = 16900 $

From this, we can find:
- $ h^2 = 22500 - c^2 $
- $ h^2 = 16900 - d^2 $

So:
$$
22500 - c^2 = 16900 - d^2 \Rightarrow d^2 - c^2 = -5600
$$
But from earlier:
$$
d^2 + c^2 = 19600
$$
Solving these equations simultaneously:

Add and subtract:
- $ 2d^2 = 19600 - 5600 = 14000 \Rightarrow d^2 = 7000 $
- $ 2c^2 = 19600 + 5600 = 25200 \Rightarrow c^2 = 12600 $
- Then $ h^2 = 22500 - 12600 = 9900 \Rightarrow h = \sqrt{9900} = 99.499 $ meters

---

### Step 2: Optimal Location for $ P $ on Segment $ CD $

We want to minimize the length of rope $ HP $, where $ P $ is a point along segment $ CD $. The rope $ HC $ and $ HD $ together have length $ 150 + 130 = 280 $ meters. Our goal is to find the **minimum** possible length of $ HP $, and then compute the **maximum savings** as:

$$
\text{Savings} = 280 - \text{min}(HP)
$$

We use geometry to find the minimal distance from $ H $ to the line segment $ CD $.

---

### Step 3: Minimal Distance from $ H $ to Line $ CD $

Let’s compute the **shortest distance from point $ H $ to the line $ CD $**.

We can use the formula for the distance from a point to a line in 3D:

$$
\text{Distance} = \frac{|(\vec{u} \times \vec{v})|}{|\vec{v}|}
$$

Where:

- $ \vec{u} = \vec{HC} = (0, -c, -h) $
- $ \vec{v} = \vec{CD} = (d, c, 0) $

So the cross product $ \vec{u} \times \vec{v} $ is:

$$
\vec{u} \times \vec{v} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & -c & -h \\
d & c & 0
\end{vmatrix}
= (hc, -hd, cd)
$$
$$
|\vec{u} \times \vec{v}| = \sqrt{(hc)^2 + (hd)^2 + (cd)^2}
$$

$$
|\vec{v}| = \sqrt{d^2 + c^2} = \sqrt{19600} = 140
$$

So the distance from $ H $ to the line $ CD $ is:

$$
\text{Distance} = \frac{\sqrt{c^2h^2 + d^2h^2 + c^2d^2}}{140}
$$

Now plug in the known values:
- $ c^2 = 12600 $
- $ d^2 = 7000 $
- $ h^2 = 9900 $
- $ c^2d^2 = 12600 \cdot 7000 = 88{,}200{,}000 $
- $ c^2h^2 = 12600 \cdot 9900 = 124{,}740{,}000 $
- $ d^2h^2 = 7000 \cdot 9900 = 69{,}300{,}000 $

So:

$$
c^2h^2 + d^2h^2 + c^2d^2 = 124{,}740{,}000 + 69{,}300{,}000 + 88{,}200{,}000 = 282{,}240{,}000
$$

$$
\text{Distance} = \frac{\sqrt{282{,}240{,}000}}{140} = \frac{16{,}800}{140} = 120 \text{ meters}
$$

---

### Step 4: Compute the Savings

The total length of the two ropes $ HC $ and $ HD $ is:

$$
HC + HD = 150 + 130 = 280 \text{ meters}
$$

The optimal single rope $ HP $ can be as short as **120 meters**, so the **maximum savings** is:

$$
280 - 120 = \boxed{160}
$$

---

### Final Answer

$$
\boxed{160}
$$

Solution:
To save the most rope, we must have $HP$ having minimum length.
For $HP$ to have minimum length, $HP$ must be perpendicular to $CD$. [asy]
pair C, D, H, P;
H=(90,120);
C=(0,0);
D=(140,0);
P=(90,0);
draw(H--C--D--H--P);
label("H", H, N);
label("C", C, SW);
label("D", D, SE);
label("P", P, S);
label("150", (C+H)/2, NW);
label("130", (D+H)/2, NE);
[/asy] (Among other things, we can see from this diagram that sliding $P$ away from the perpendicular position does make $HP$ longer.)
In the diagram, $HC=150$, $HD=130$ and $CD=140$.
Let $HP=x$ and $PD=a$.  Then $CP=140-a$.
By the Pythagorean Theorem in $\triangle HPC$, $x^2 + (140-a)^2 = 150^2$.
By the Pythagorean Theorem in $\triangle HPD$, $x^2+a^2 = 130^2$.
Subtracting the second equation from the first, we obtain \begin{align*}
(140-a)^2 - a^2 & = 150^2 - 130^2 \\
(19600 - 280a+a^2)-a^2 & = 5600 \\
19600 -280a & = 5600 \\
280a & = 14000 \\
a & = 50
\end{align*} Therefore, $x^2 + 90^2 = 150^2$ or $x^2 = 150^2 - 90^2 = 22500 - 8100 = 14400$ so $x =120$.
So the shortest possible rope that we can use is 120 m, which saves $130+150-120 = \boxed{160}$ m of rope.