Explanation:
Okay, let me try to figure out how to solve this problem. The question is asking for the minimum value of the expression:

$$
\sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{x^2 + y^2 - 80x - 100y + 4100}
$$

given that $0 \leq x \leq 40$ and $0 \leq y \leq 50$.

First, I need to understand what each term represents. The first two terms look like distances. For example, $\sqrt{x^2 + 400}$ can be thought of as the distance from the point $(x, 0)$ to the point $(0, 20)$ since $\sqrt{(x - 0)^2 + (0 - 20)^2} = \sqrt{x^2 + 400}$. Similarly, $\sqrt{y^2 + 900}$ is the distance from $(0, y)$ to $(0, 30)$ because $\sqrt{(0 - 0)^2 + (y - 30)^2} = \sqrt{y^2 + 900}$. Wait, actually, if we consider coordinates, maybe we need to think differently. Let me check:

If we have a general distance formula between two points $(a, b)$ and $(c, d)$, it's $\sqrt{(a - c)^2 + (b - d)^2}$. So for the first term, if I set $x$ as the x-coordinate and the other term as the y-coordinate, but there's no y in the first term. Maybe I need to think of it as distances from some fixed points.

Alternatively, perhaps all three terms can be represented as distances in some coordinate system. Let me think.

Let me write the third term:

$$
\sqrt{x^2 + y^2 - 80x - 100y + 4100}
$$

Let me simplify the expression inside the square root:

$x^2 + y^2 - 80x - 100y + 4100$

I can complete the squares for x and y terms.

For the x terms: $x^2 - 80x$. Completing the square:

$x^2 - 80x = (x - 40)^2 - 1600$

Similarly for y terms: $y^2 - 100y = (y - 50)^2 - 2500$

So substituting back into the expression:

$$
(x - 40)^2 - 1600 + (y - 50)^2 - 2500 + 4100
$$

Calculating constants: -1600 -2500 + 4100 = (-4100) + 4100 = 0.

Therefore, the third term simplifies to:

$$
\sqrt{(x - 40)^2 + (y - 50)^2}
$$

Oh! That's interesting. So the third term is the distance between the point $(x, y)$ and the point $(40, 50)$.

So now, the entire expression is the sum of three distances:

1. $\sqrt{x^2 + 400}$: distance from $(x, 0)$ to $(0, 20)$? Wait, no. Wait, if we think of it as a distance between two points, but the first term is $\sqrt{x^2 + 400}$. Let me check again. Let me imagine that we are dealing with coordinates.

Suppose we have three points:

Let me think of this as a problem in geometry. Let me try to interpret each term as a distance.

First term: $\sqrt{x^2 + 20^2}$, so this is the distance from the point (x, 0) to (0, 20)? Wait, no. Wait, if you have a point (x, y), then the distance from (x, y) to (a, b) is sqrt[(x - a)^2 + (y - b)^2]. So if our first term is sqrt[x^2 + 400], that's sqrt[x^2 + 20^2], which would be the distance from (x, 0) to (0, 20)? Wait, but if we have (x, 0) and (0, 20), then the distance is sqrt[(x - 0)^2 + (0 - 20)^2] = sqrt(x^2 + 400). So yes, that's correct. Similarly, sqrt[y^2 + 900] is sqrt[y^2 + 30^2], so distance from (0, y) to (0, 30)? Wait, but (0, y) to (0, 30) would be |y - 30|, but sqrt(y^2 + 900) is not |y - 30|. Wait, maybe I need a different approach.

Alternatively, maybe all three terms are distances from points in some coordinate system. Let me try to think of the problem as minimizing the sum of three distances. Let me consider points in 2D space.

Suppose we have three points:

Let me define points A, B, and C such that:

- The first term is the distance from point P(x, y) to point A.

- The second term is the distance from point P(x, y) to point B.

- The third term is the distance from point P(x, y) to point C.

Wait, but the first two terms don't involve both x and y. Wait, first term is sqrt(x² + 400), which is sqrt((x - 0)^2 + (0 - 20)^2) if we think of a point (x, 0) and (0, 20). Similarly, sqrt(y² + 900) is sqrt((0 - 0)^2 + (y - 30)^2) if we consider (0, y) and (0, 30). But if we want to think of a single point (x, y), then sqrt(x² + 400) would be the distance from (x, y) to (0, 20) only if the y-coordinate is zero. But since in the problem, the variables x and y are independent, maybe this approach isn't directly applicable.

Alternatively, maybe the problem involves reflecting points or something like that. Wait, let me think again. Since the third term is the distance from (x, y) to (40, 50). Let me denote the third term as distance from (x, y) to D(40, 50).

Now, the first term sqrt(x² + 400) is the distance from (x, 0) to (0, 20). But if we think of it as a single point, maybe we need to consider another coordinate system. Alternatively, maybe think of it as the distance from (x, 0) to (0, 20) and the distance from (0, y) to (0, 30), but then how do they relate to the point (x, y)? Maybe the problem involves multiple points?

Alternatively, perhaps we need to model this as a path. Suppose we have a path that goes from (0, 20) to (x, 0) to (x, y) to (40, 50) or something like that? Wait, that seems complicated. Let me try to think of the total expression as sum of three distances.

Wait, if we consider three points:

Let me define:

- Point A at (0, 20)

- Point B at (0, 30)

- Point C at (40, 50)

Then the expression is:

sqrt[(x - 0)^2 + (0 - 20)^2] + sqrt[(y - 0)^2 + (0 - 30)^2] + sqrt[(x - 40)^2 + (y - 50)^2]

Wait, but that's not exactly. Wait, sqrt[x² + 400] is sqrt[(x - 0)^2 + (0 - 20)^2], so that's the distance between (x, 0) and (0, 20). Similarly, sqrt[y² + 900] is the distance between (0, y) and (0, 30). But then we have the distance between (x, y) and (40, 50).

But how do these relate? Maybe the variables x and y are coordinates of some point, and we have to consider distances from different points. However, the problem is that the first two terms have (x, 0) and (0, y), which are points on the axes, but the third term is from (x, y).

Alternatively, maybe think of this as a path: starting at (0, 20), going to (x, 0), then to (x, y), then to (40, 50)? But then that would be the sum of three distances, but the second term is from (x, 0) to (0, y)? Not sure. Maybe another way.

Alternatively, maybe there's a way to interpret all three terms as distances from the same point. Let me think again.

Alternatively, think of the entire expression as three distances that need to be minimized. Let me suppose that x and y are variables. So we need to minimize the sum of three distances: the distance from (x, 0) to (0, 20), the distance from (0, y) to (0, 30), and the distance from (x, y) to (40, 50). But how do these relate?

Alternatively, maybe the problem is about finding a point (x, y) such that the sum of three specific distances is minimized. Wait, if I consider (x, y) as a variable point, then:

- sqrt(x² + 400) is the distance from (x, y) to (0, 20) only if y is fixed? Wait, no. Wait, if (x, y) is a variable point, then sqrt(x² + 400) is not necessarily a distance to a fixed point unless we fix y. So perhaps there's a misinterpretation here. Maybe we need to model this differently.

Alternatively, maybe consider coordinates. Let me think about each term:

First term: sqrt(x² + 20²). If we consider that as the distance from (x, 0) to (0, 20), but then (x, 0) is a point on the x-axis. Similarly, sqrt(y² + 30²) is the distance from (0, y) to (0, 30), so that's a point on the y-axis. Then the third term is the distance between (x, y) and (40, 50).

But how do these terms relate? If we imagine that there is a path that goes from (0, 20) to (x, 0) to (x, y) to (40, 50), but that seems complicated. Maybe we need to consider reflections?

Alternatively, maybe there's a geometric interpretation where we can use the reflection principle to find minimal paths. For instance, in problems where you have to find minimal paths that reflect off a line, you can reflect a point across the line and find a straight line distance.

Wait, here's an idea. Let me think of the entire expression as the sum of three distances:

Let me consider points:

- Let’s define point P = (x, 0), Q = (0, y), and R = (x, y). However, that might not directly help.

Alternatively, perhaps think of the expression as the sum of three distances:

From (x, 0) to (0, 20) [sqrt(x² + 400)]

Plus the distance from (0, y) to (0, 30) [sqrt(y² + 900)]

Plus the distance from (x, y) to (40, 50) [sqrt((x - 40)^2 + (y - 50)^2)]

But how do these three terms relate? If we need to minimize this sum with respect to x and y? It seems that x and y are variables, but each term has different dependencies. However, maybe there is a way to relate them.

Alternatively, think of the problem as optimizing over x and y. Maybe take partial derivatives and set them to zero to find critical points? Although that might be complicated with square roots, but maybe manageable.

Alternatively, maybe there's a way to combine the terms into a single expression. Let me try to think geometrically again. Let me consider that the first term is distance from (x, 0) to (0, 20), the second term is distance from (0, y) to (0, 30), and the third term is distance from (x, y) to (40, 50). If we imagine that x and y are coordinates of a point (x, y), then maybe there are some relationships between these points.

Wait, perhaps if we consider that (x, 0) and (0, y) are projections of the point (x, y) onto the axes? Maybe not.

Alternatively, suppose that we have to go from (0, 20) to (x, 0) to (x, y) to (40, 50), and also have to go from (0, y) to (0, 30)? Not sure. This seems too vague.

Alternatively, maybe consider that the total expression is the sum of:

- Distance from (x, 0) to (0, 20)

- Distance from (0, y) to (0, 30)

- Distance from (x, y) to (40, 50)

But since x and y are variables, maybe the minimum occurs when these points are aligned in some way? For example, if we could have (x, 0) connected to (0, 20), (0, y) connected to (0, 30), and (x, y) connected to (40, 50), but I don't see a direct connection.

Alternatively, perhaps think of the entire expression as a combination of distances from different points. Let me think if there is a way to represent all three terms as parts of a single path.

Wait, here's a different idea. Let me define three points:

Let’s say we have a point A at (0, 20), point B at (0, 30), and point C at (40, 50). Now, suppose that we have a point P = (x, 0) and a point Q = (0, y). Then, maybe the total expression is the sum of distance from A to P, distance from B to Q, and distance from P to Q? Wait, no. Wait, distance from A to P is sqrt(x² + (0 - 20)^2) = sqrt(x² + 400), which matches the first term. Similarly, distance from B to Q is sqrt((0 - 0)^2 + (y - 30)^2) = sqrt(y² + 900), which matches the second term. Then the distance from P to Q is sqrt((x - 0)^2 + (0 - y)^2) = sqrt(x² + y²). But the third term is sqrt(x² + y² - 80x - 100y + 4100). Which we earlier simplified to sqrt((x - 40)^2 + (y - 50)^2). So that doesn't match.

Wait, but the third term is actually the distance from (x, y) to (40, 50). So if I consider another point, maybe D = (40, 50), then the third term is the distance between (x, y) and D. But how does that relate to the other terms?

Alternatively, maybe if I consider all three terms as distances in a coordinate system. Let me try to visualize.

Suppose we have a path that goes from point A (0, 20) to point P (x, 0), then to point Q (0, y), then to point D (40, 50). Then, the total distance would be:

Distance AP + PQ + QD. Where AP is sqrt(x² + 400), PQ is sqrt(x² + y²), and QD is sqrt((0 - 40)^2 + (y - 50)^2) = sqrt(1600 + (y - 50)^2). But that's different from our original expression, which has the third term as distance from (x, y) to (40, 50). So it's not quite matching.

Alternatively, maybe think of the expression as AP + BQ + PD, where PD is the distance from P to D. But then, PD would be sqrt((x - 40)^2 + (0 - 50)^2) = sqrt((x - 40)^2 + 2500), which is different from the third term.

Alternatively, perhaps we need to think of the third term as the distance between (x, y) and (40, 50), so if we consider a point (x, y), then the total expression is:

Distance from (x, 0) to (0, 20) + Distance from (0, y) to (0, 30) + Distance from (x, y) to (40, 50)

But how do these connect? If we imagine that (x, 0) and (0, y) are variable points, and (x, y) is another variable point? But this seems too vague.

Alternatively, maybe there's a way to relate x and y through some constraints? For example, if we consider that (x, 0) and (0, y) are connected to (x, y), but again, not sure.

Wait, going back, maybe the key is in the third term, which is the distance between (x, y) and (40, 50). So the entire expression is:

sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)^2 + (y - 50)^2)

But perhaps I need to see this as a sum of three distances from three different points to some variable point? But no, the first two terms are not distances from (x, y) to fixed points.

Wait, unless we interpret them as distances from (x, 0) to (0, 20) and from (0, y) to (0, 30). If we think of x and y as coordinates, maybe we need to find a point (x, y) such that...?

Alternatively, maybe think of this as a problem in optimization with two variables. Let me consider trying calculus here. Let me denote the function as:

f(x, y) = sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)² + (y - 50)²)

We need to find the minimum of f(x, y) over 0 ≤ x ≤ 40 and 0 ≤ y ≤ 50.

This seems like a problem that could be approached with partial derivatives. Let's try that.

First, compute the partial derivatives of f with respect to x and y, set them to zero, and solve the equations.

Let me denote:

f(x, y) = sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)² + (y - 50)²)

Let me compute ∂f/∂x:

The derivative of sqrt(x² + 400) with respect to x is (x)/sqrt(x² + 400).

The derivative of sqrt(y² + 900) with respect to x is 0.

The derivative of sqrt((x - 40)^2 + (y - 50)^2) with respect to x is [(x - 40)] / sqrt((x - 40)^2 + (y - 50)^2).

Therefore,

∂f/∂x = [x / sqrt(x² + 400)] + [(x - 40) / sqrt((x - 40)^2 + (y - 50)^2)].

Similarly, compute ∂f/∂y:

Derivative of sqrt(x² + 400) with respect to y is 0.

Derivative of sqrt(y² + 900) with respect to y is [y / sqrt(y² + 900)].

Derivative of sqrt((x - 40)^2 + (y - 50)^2) with respect to y is [(y - 50) / sqrt((x - 40)^2 + (y - 50)^2)].

Therefore,

∂f/∂y = [y / sqrt(y² + 900)] + [(y - 50) / sqrt((x - 40)^2 + (y - 50)^2)].

To find critical points, set ∂f/∂x = 0 and ∂f/∂y = 0.

So,

(1) x / sqrt(x² + 400) + (x - 40) / sqrt((x - 40)^2 + (y - 50)^2) = 0

(2) y / sqrt(y² + 900) + (y - 50) / sqrt((x - 40)^2 + (y - 50)^2) = 0

This seems quite complex. Maybe there's a way to relate equations (1) and (2). Let me denote:

Let me call the third term's square root as D = sqrt((x - 40)^2 + (y - 50)^2).

Then equations (1) and (2) become:

(1) x / sqrt(x² + 400) + (x - 40)/D = 0

(2) y / sqrt(y² + 900) + (y - 50)/D = 0

Let me rearrange equation (1):

(x)/sqrt(x² + 400) = -(x - 40)/D

Similarly, equation (2):

(y)/sqrt(y² + 900) = -(y - 50)/D

So, from (1):

Let me square both sides of equation (1):

[x² / (x² + 400)] = [(x - 40)^2]/D²

Similarly for equation (2):

[y² / (y² + 900)] = [(y - 50)^2]/D²

But D² is (x - 40)^2 + (y - 50)^2. Let me denote D² = (x - 40)^2 + (y - 50)^2.

So from equation (1):

x² / (x² + 400) = (x - 40)^2 / [(x - 40)^2 + (y - 50)^2]

Similarly, from equation (2):

y² / (y² + 900) = (y - 50)^2 / [(x - 40)^2 + (y - 50)^2]

Let me denote S = (x - 40)^2 + (y - 50)^2.

Then:

From equation (1): x² / (x² + 400) = (x - 40)^2 / S

From equation (2): y² / (y² + 900) = (y - 50)^2 / S

Let me solve for S from both equations and set them equal.

From equation (1):

S = [ (x - 40)^2 * (x² + 400) ] / x²

From equation (2):

S = [ (y - 50)^2 * (y² + 900) ] / y²

Therefore, setting them equal:

[ (x - 40)^2 * (x² + 400) ] / x² = [ (y - 50)^2 * (y² + 900) ] / y²

This seems very complicated. Maybe there's a relationship between x and y that we can exploit?

Alternatively, maybe assume that there is some proportionality between the terms. Let me think. If I denote the first fraction as something and second as something else. Alternatively, notice that 400 is 20² and 900 is 30², so maybe there's a connection with the other points.

Wait, let's think geometrically again. Let me consider the points mentioned:

- The first term is distance from (x, 0) to (0, 20)

- The second term is distance from (0, y) to (0, 30)

- The third term is distance from (x, y) to (40, 50)

But if I imagine that (x, 0) is a point on the x-axis, and (0, y) is a point on the y-axis, and (x, y) is some point in the plane. However, how do these relate?

Alternatively, maybe think of this problem as optimizing over x and y where x and y are coordinates of a point, and we have three different distances. But maybe there's a way to combine them into a single path.

Wait, here's an idea. Let me imagine that we need to find a point (x, y) such that the total expression is minimized. Let me think of the first term sqrt(x² + 400) as the distance from (x, y) to (0, 20) if y is fixed? No. Wait, if we consider the first term sqrt(x² + 400) as sqrt((x - 0)^2 + (y - 20)^2), but that would be if y is 0. Wait, no. If we set y = 0, then it would be sqrt(x² + 400). So maybe the first term is the distance from (x, 0) to (0, 20). Similarly, the second term sqrt(y² + 900) is the distance from (0, y) to (0, 30).

But if we think of x and y as coordinates such that the point (x, 0) is connected to (0, 20), and (0, y) is connected to (0, 30), and then the point (x, y) is connected to (40, 50). But how do these connect?

Alternatively, maybe think of the entire expression as the sum of three distances:

From (0, 20) to (x, 0) to (40, 50) and also from (0, 30) to (0, y) to (x, y)? This is too vague.

Alternatively, maybe think of it as three separate distances, each with their own constraints. Wait, perhaps there is a way to interpret this as minimizing the sum of three distances where the points are connected in a certain way. For example, imagine that we have to go from point A (0, 20) to some point on the x-axis (x, 0), then from there to a point (x, y), then from (x, y) to point D (40, 50), and also go from point B (0, 30) to some point on the y-axis (0, y), and then from there to (x, y). But this seems too convoluted.

Alternatively, perhaps there is an optimal point (x, y) that relates to reflections. For example, in problems where you have to find minimal paths that reflect off a surface, you can reflect points across the surface.

Let me try to think of this as a path: suppose we have to go from point A to point P on the x-axis, then from P to point Q, then from Q to point D, and also from point B to point Q on the y-axis?

Wait, here's another thought: If I consider that sqrt(x² + 400) is the distance from A(0,20) to P(x, 0), sqrt(y² + 900) is the distance from B(0,30) to Q(0, y), and sqrt((x - 40)^2 + (y - 50)^2) is the distance from Q(x, y) to D(40, 50). Wait, but Q(0, y) and Q(x, y) are different. Maybe not.

Alternatively, if I consider that there is a path from A(0,20) to P(x, 0) to D(40, 50), which would be sqrt(x² + 400) + sqrt((x - 40)^2 + (0 - 50)^2). But our third term is sqrt((x - 40)^2 + (y - 50)^2). So this is similar but with y.

Alternatively, maybe combine all three terms as paths from A to P to Q to D and B to Q? Not sure.

Alternatively, think of the entire expression as the sum of distances from A to P, B to Q, and Q to D, where P is (x, 0) and Q is (x, y). Wait, then that would be sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)^2 + (y - 50)^2). But then Q is (x, y), so if we have to go from B(0, 30) to Q(0, y)? Wait, no. If Q is (x, y), then distance from B(0, 30) to Q(x, y) is sqrt(x² + (y - 30)^2). But in our expression, the second term is sqrt(y² + 900) = sqrt(y² + 30²), which is the distance from (0, y) to (0, 30). So that's a different point Q on the y-axis.

This is getting too tangled. Maybe I need to step back and think of specific values.

Let me check if the minimum occurs at some boundary points. Since x and y are constrained between 0 and 40, 0 and 50, respectively, maybe the minimum is achieved at one of the corners? For example, check at (0, 0), (40, 0), (0, 50), (40, 50), or maybe somewhere in the middle.

But before that, let me think if there's a geometric interpretation. Suppose the expression represents the sum of three distances:

- From (x, 0) to (0, 20),

- From (0, y) to (0, 30),

- From (x, y) to (40, 50).

If we imagine that (x, 0) and (0, y) are points on the axes, and (x, y) is a point. Maybe there's a relation between them. For example, if we think of the total expression as the sum of distances from (x, 0) to (0, 20), (0, y) to (0, 30), and (x, y) to (40, 50). But how?

Alternatively, think of it as:

Total distance = distance from A(0,20) to P(x,0) + distance from B(0,30) to Q(0,y) + distance from R(x,y) to D(40,50). If P, Q, R are related somehow? If P(x,0), Q(0,y), R(x,y), then R is the point (x,y). So maybe it's like the sum of distances from A to P, B to Q, and R to D, where P is on the x-axis, Q is on the y-axis, and R is the point (x,y). But how do these connect?

Alternatively, maybe think of the entire expression as the sum of distances from three different points to other points, but without a clear relationship.

Alternatively, maybe think about the following: if we fix x and y, then the expression is the sum of these three terms. Maybe there is a way to relate all three terms through coordinates.

Wait, going back to the third term, which simplifies to sqrt((x - 40)^2 + (y - 50)^2). Let me call this distance from (x, y) to (40, 50). Let me call this point E(40, 50). So the third term is distance from (x, y) to E.

Now, first term sqrt(x² + 400) is distance from (x, 0) to (0, 20). Let me call this point F(0, 20). Similarly, the second term sqrt(y² + 900) is distance from (0, y) to (0, 30). Let me call this point G(0, 30).

So the total expression is:

distance from F(0, 20) to P(x, 0) + distance from G(0, 30) to Q(0, y) + distance from R(x, y) to E(40, 50).

But how are P, Q, R related?

If I consider that P is (x, 0), Q is (0, y), and R is (x, y). So R is connected to P and Q. Maybe think of this as triangles or something. But I don't see the connection.

Alternatively, think of this as minimizing the sum of distances from F to P, from G to Q, and from R to E, where P is on x-axis, Q is on y-axis, and R is some point in the plane. But without a direct relation between P, Q, R, it's unclear.

Alternatively, maybe think about optimizing x and y such that the three terms are minimized. For instance, if we can find x and y such that the three terms have some sort of equilibrium.

Alternatively, going back to the calculus approach. Let me try to see if I can solve the equations:

From equations (1) and (2), we had:

x / sqrt(x² + 400) = -(x - 40)/D

and

y / sqrt(y² + 900) = -(y - 50)/D

Where D = sqrt((x - 40)^2 + (y - 50)^2)

Let me denote:

Let me define:

Let me set u = x / sqrt(x² + 400)

Let me set v = y / sqrt(y² + 900)

Then from equations:

u = -(x - 40)/D

v = -(y - 50)/D

So, multiplying both equations:

u * v = [(x - 40)(y - 50)] / D²

But D² = (x - 40)^2 + (y - 50)^2

So:

u * v = [(x - 40)(y - 50)] / [(x - 40)^2 + (y - 50)^2]

Hmm.

Also, note that u = x / sqrt(x² + 400) implies that u is the cosine of the angle between the vector (x, 0) and the vector to (0, 20)? Wait, not sure.

Alternatively, perhaps express u and v in terms of angles. For example, for u:

If I imagine a right triangle with base x and height 20, then u = x / sqrt(x² + 20²) = cos(theta), where theta is the angle between the hypotenuse and the x-axis. Similarly, v = y / sqrt(y² + 30²) = cos(phi), where phi is the angle between hypotenuse and y-axis.

But not sure if helpful.

Alternatively, note that from u = -(x - 40)/D and v = -(y - 50)/D, we can write:

Let me think of D as the hypotenuse of a right triangle with legs (x - 40) and (y - 50). So, u = - (x - 40)/D and v = - (y - 50)/D. So u and v are the x and y components of the unit vector pointing from (x, y) to (40, 50), but with a negative sign.

Wait, if we consider the vector from (x, y) to (40, 50), its components are (40 - x, 50 - y). The unit vector in that direction is [(40 - x)/D, (50 - y)/D]. So, according to our previous equations, u = - (x - 40)/D = (40 - x)/D, and similarly v = - (y - 50)/D = (50 - y)/D. Therefore, u and v are the components of the unit vector from (x, y) to (40, 50).

But u was defined as x / sqrt(x² + 400), which is the cosine of the angle made by the vector (x, 0) with the x-axis. Similarly, v is y / sqrt(y² + 900), which is the cosine of the angle made by (0, y) with the y-axis.

Therefore, we have:

(40 - x)/D = u = x / sqrt(x² + 400)

(50 - y)/D = v = y / sqrt(y² + 900)

Therefore:

(40 - x)/D = x / sqrt(x² + 400)

(50 - y)/D = y / sqrt(y² + 900)

Let me square both equations:

For the first equation:

[(40 - x)^2]/D² = x² / (x² + 400)

Similarly, second equation:

[(50 - y)^2]/D² = y² / (y² + 900)

But D² = (x - 40)^2 + (y - 50)^2 = (40 - x)^2 + (50 - y)^2

So from first equation:

[(40 - x)^2] / [(40 - x)^2 + (50 - y)^2] = x² / (x² + 400)

Similarly, second equation:

[(50 - y)^2] / [(40 - x)^2 + (50 - y)^2] = y² / (y² + 900)

Let me denote A = (40 - x)^2 and B = (50 - y)^2. Then D² = A + B.

So first equation:

A / (A + B) = x² / (x² + 400)

Second equation:

B / (A + B) = y² / (y² + 900)

Let me call these equations (1) and (2).

Let me denote the left-hand sides as fractions:

From equation (1): A / (A + B) = x² / (x² + 400)

From equation (2): B / (A + B) = y² / (y² + 900)

Let me add both equations:

[A / (A + B)] + [B / (A + B)] = [x² / (x² + 400)] + [y² / (y² + 900)]

Which simplifies to:

1 = [x² / (x² + 400)] + [y² / (y² + 900)]

So, this gives us an equation relating x and y:

x² / (x² + 400) + y² / (y² + 900) = 1

Let me write this as:

Let me denote u = x² and v = y². Then the equation becomes:

u / (u + 400) + v / (v + 900) = 1

Solving for u and v.

Let me solve this equation:

u / (u + 400) + v / (v + 900) = 1

Let me rearrange:

u / (u + 400) = 1 - v / (v + 900)

Compute the right side:

1 - v / (v + 900) = [(v + 900) - v] / (v + 900) = 900 / (v + 900)

Therefore:

u / (u + 400) = 900 / (v + 900)

Cross-multiplying:

u(v + 900) = 900(u + 400)

Expanding:

uv + 900u = 900u + 360000

Subtract 900u from both sides:

uv = 360000

Thus, uv = 360000

So, x² * y² = 360000

Therefore, xy = ±600. But since x and y are non-negative (between 0 and 40 for x, 0 and 50 for y), xy = 600.

So, we have that xy = 600.

Now, this is an important relation: x * y = 600.

So, we have that x * y = 600.

Now, remember that we also have from equations (1) and (2):

From equation (1): A / (A + B) = x² / (x² + 400), where A = (40 - x)^2 and B = (50 - y)^2.

But now, with xy = 600, perhaps we can find relations between x and y.

So, let me try to relate x and y using xy = 600. Since x is between 0 and 40, and y is between 0 and 50, let me check if there are possible solutions.

First, x must satisfy 0 ≤ x ≤ 40, so y = 600 / x must satisfy 0 ≤ y ≤ 50.

So, y = 600/x ≤ 50 => 600/x ≤ 50 => x ≥ 600 / 50 = 12.

Also, since x ≥ 0, and y ≥ 0, we have that x must be between 12 and 40. Because if x is between 0 and 12, y would be greater than 50, which is outside the allowed range. Therefore, x ∈ [12, 40], and y = 600 / x ∈ [12, 50] (since when x=40, y=15; when x=12, y=50). So that fits within the given constraints 0 ≤ y ≤ 50.

So, we have x ∈ [12, 40], y = 600/x.

Now, with this relation, we can substitute into one of our previous equations. Let me recall that from equation (1):

A / (A + B) = x² / (x² + 400)

But with A = (40 - x)^2 and B = (50 - y)^2 = (50 - 600/x)^2.

So, substituting:

[(40 - x)^2] / [(40 - x)^2 + (50 - 600/x)^2] = x² / (x² + 400)

This seems complicated, but maybe we can solve it.

Let me denote t = x for simplicity. Then y = 600/t.

So, equation becomes:

[(40 - t)^2] / [(40 - t)^2 + (50 - 600/t)^2] = t² / (t² + 400)

Let me cross-multiplying:

(40 - t)^2 * (t² + 400) = t² * [(40 - t)^2 + (50 - 600/t)^2]

Let me expand both sides.

First, left side:

(40 - t)^2 * (t² + 400)

Right side:

t² * [(40 - t)^2 + (50 - 600/t)^2]

Let me compute the right side first:

First compute (50 - 600/t)^2:

= (50 - 600/t)^2 = [(50t - 600)/t]^2 = (50t - 600)^2 / t²

Therefore, the right side becomes:

t² * [(40 - t)^2 + (50t - 600)^2 / t²] = t²*(40 - t)^2 + (50t - 600)^2

So, the equation is:

Left side: (40 - t)^2 * (t² + 400)

Right side: t²*(40 - t)^2 + (50t - 600)^2

Subtract right side from left side:

(40 - t)^2 * (t² + 400) - [t²*(40 - t)^2 + (50t - 600)^2] = 0

Simplify:

(40 - t)^2 * (t² + 400) - t²*(40 - t)^2 - (50t - 600)^2 = 0

Factor out (40 - t)^2:

(40 - t)^2 [t² + 400 - t²] - (50t - 600)^2 = 0

Simplify inside the brackets:

t² + 400 - t² = 400

Therefore:

(40 - t)^2 * 400 - (50t - 600)^2 = 0

Expand both terms:

First term: 400*(40 - t)^2

Second term: (50t - 600)^2

So:

400*(40 - t)^2 - (50t - 600)^2 = 0

Let me expand both terms.

First, expand 400*(40 - t)^2:

= 400*(1600 - 80t + t²) = 400*1600 - 400*80t + 400*t² = 640000 - 32000t + 400t²

Second, expand (50t - 600)^2:

= (50t)^2 - 2*50t*600 + 600² = 2500t² - 60000t + 360000

Therefore, the equation becomes:

[640000 - 32000t + 400t²] - [2500t² - 60000t + 360000] = 0

Simplify:

640000 - 32000t + 400t² - 2500t² + 60000t - 360000 = 0

Combine like terms:

t² terms: 400t² - 2500t² = -2100t²

t terms: -32000t + 60000t = 28000t

Constants: 640000 - 360000 = 280000

Thus, equation:

-2100t² + 28000t + 280000 = 0

Divide both sides by -100 to simplify:

21t² - 280t - 2800 = 0

Wait, let me check:

-2100t² / -100 = 21t²

28000t / -100 = -280t

280000 / -100 = -2800

Wait, but the original equation was:

-2100t² + 28000t + 280000 = 0

Dividing by -100:

21t² - 280t - 2800 = 0

Yes.

So we have quadratic equation:

21t² - 280t - 2800 = 0

Let me simplify this equation by dividing by common factors. Let me check GCD of coefficients 21, 280, 2800.

Factors of 21: 3, 7

Factors of 280: 2^3 * 5 * 7

Factors of 2800: 2^4 * 5^2 * 7

So common factor is 7.

Divide by 7:

3t² - 40t - 400 = 0

Now solve 3t² - 40t - 400 = 0.

Use quadratic formula: t = [40 ± sqrt(1600 + 4800)] / 6

Because discriminant D = (-40)^2 - 4*3*(-400) = 1600 + 4800 = 6400

sqrt(6400) = 80

Therefore, t = [40 ± 80] / 6

So two solutions:

t = (40 + 80)/6 = 120/6 = 20

t = (40 - 80)/6 = -40/6 = -20/3 ≈ -6.666...

But since t = x ∈ [12, 40], we discard the negative solution. Therefore, t = 20.

Thus, x = 20. Then y = 600 / x = 600 / 20 = 30.

So the critical point is at (x, y) = (20, 30).

Now we need to check if this is a minimum. Also, since our domain is 0 ≤ x ≤ 40 and 0 ≤ y ≤ 50, and we found that x must be between 12 and 40, and y between 12 and 50, this point (20, 30) lies within the domain.

Now, we need to verify if this critical point gives a minimum. Also, we should check the boundaries since the minimum could occur there.

But first, let's compute the value of the function at (20, 30):

Compute each term:

First term: sqrt(x² + 400) = sqrt(20² + 20²) = sqrt(400 + 400) = sqrt(800) = 20*sqrt(2) ≈ 28.284

Second term: sqrt(y² + 900) = sqrt(30² + 30²) = sqrt(900 + 900) = sqrt(1800) = 30*sqrt(2) ≈ 42.426

Third term: sqrt((x - 40)^2 + (y - 50)^2) = sqrt((20 - 40)^2 + (30 - 50)^2) = sqrt((-20)^2 + (-20)^2) = sqrt(400 + 400) = sqrt(800) = 20*sqrt(2) ≈ 28.284

Total sum: 20√2 + 30√2 + 20√2 = 70√2 ≈ 70 * 1.4142 ≈ 98.994

Now, let's check the boundaries. Since the minimum could be on the boundary of the domain, we need to check some corner points and edges.

First, let's check the corners:

Point (0, 0):

First term: sqrt(0 + 400) = 20

Second term: sqrt(0 + 900) = 30

Third term: sqrt((0 - 40)^2 + (0 - 50)^2) = sqrt(1600 + 2500) = sqrt(4100) ≈ 64.03

Total ≈ 20 + 30 + 64.03 ≈ 114.03

Which is greater than 70√2 ≈ 98.99.

Point (40, 50):

First term: sqrt(40² + 400) = sqrt(1600 + 400) = sqrt(2000) ≈ 44.721

Second term: sqrt(50² + 900) = sqrt(2500 + 900) = sqrt(3400) ≈ 58.309

Third term: sqrt((40 - 40)^2 + (50 - 50)^2) = 0

Total ≈ 44.721 + 58.309 + 0 ≈ 103.03, which is still higher than ≈98.99.

Another corner: (40, 0):

First term: sqrt(40² + 400) = sqrt(1600 + 400) = sqrt(2000) ≈ 44.721

Second term: sqrt(0 + 900) = 30

Third term: sqrt((40 - 40)^2 + (0 - 50)^2) = sqrt(0 + 2500) = 50

Total ≈ 44.721 + 30 + 50 ≈ 124.72, which is higher.

Similarly, (0, 50):

First term: sqrt(0 + 400) = 20

Second term: sqrt(50² + 900) = sqrt(2500 + 900) = sqrt(3400) ≈ 58.309

Third term: sqrt((0 - 40)^2 + (50 - 50)^2) = 40

Total ≈ 20 + 58.309 + 40 ≈ 118.309, which is higher.

So corners are worse. Now check edges.

First, check the edge where x = 12 (since x ≥ 12 for y ≤ 50). Wait, x ranges from 12 to 40. So maybe check if the minimal value occurs on the boundary of x=12 or y=...?

Alternatively, since we found a critical point inside the domain, and the value there is approximately 98.99, which is less than the corners, perhaps it is the minimum. But to be thorough, check some other points.

For example, check the point (20, 30): already did, value ~98.99.

Check another point on the boundary where y=50. Since y=50, then from xy=600, x=600/50=12. So (12, 50). Let's compute the value:

First term: sqrt(12² + 400) = sqrt(144 + 400) = sqrt(544) ≈ 23.32

Second term: sqrt(50² + 900) = sqrt(2500 + 900) = sqrt(3400) ≈ 58.31

Third term: sqrt((12 - 40)^2 + (50 - 50)^2) = sqrt((-28)^2 + 0) = 28

Total ≈ 23.32 + 58.31 + 28 ≈ 109.63, which is more than 98.99.

Check edge where x=40. Then y=600/40=15. So (40, 15):

First term: sqrt(40² + 400) = sqrt(1600 + 400) = sqrt(2000) ≈ 44.72

Second term: sqrt(15² + 900) = sqrt(225 + 900) = sqrt(1125) ≈ 33.54

Third term: sqrt((40 - 40)^2 + (15 - 50)^2) = sqrt(0 + (-35)^2) = 35

Total ≈ 44.72 + 33.54 + 35 ≈ 113.26, which is more than 98.99.

How about the edge where y=0. Then x=600/0 undefined, but since y cannot be 0 (since x would be infinite). But since y ≥ 0, but the minimal x for y=0 would be undefined. Hence, edge y=0 is not in our domain except at (x, 0) where x is something. Wait, maybe check point where y=0, but since from xy=600, if y=0, x is undefined. So no solution.

Similarly, for edge x=0, then y=600/0 undefined. So no solution.

Now, check another point, say, midpoint of the critical point. Wait, we already have the critical point.

Alternatively, check another point where x=20, y=30. Let me compute the value again:

First term: sqrt(20² + 20²) = sqrt(800) ≈ 28.284

Second term: sqrt(30² + 30²) = sqrt(1800) ≈ 42.426

Third term: sqrt((20-40)^2 + (30-50)^2) = sqrt(400 + 400) = sqrt(800) ≈ 28.284

Total: 28.284 + 42.426 + 28.284 ≈ 98.994, which is about 70√2 since √2 ≈ 1.4142, 70 * 1.4142 ≈ 98.994.

Now, to confirm this is indeed the minimum, let's check another point near (20, 30). Let me take x=20, y=30. Suppose we perturb a bit, say x=20 + Δx, y=30 + Δy. Let me check with Δx = 1, Δy = -1 (since xy=600, if we change x and y, need to maintain xy=600). But if we take x=21, then y=600/21 ≈ 28.571. Let's compute the value:

First term: sqrt(21² + 400) = sqrt(441 + 400) = sqrt(841) = 29

Second term: sqrt( (600/21)^2 + 900 ). Let me compute y ≈ 28.571, so y² ≈ (600/21)^2 ≈ (28.571)^2 ≈ 816.326. Then sqrt(816.326 + 900) ≈ sqrt(1716.326) ≈ 41.43

Third term: sqrt( (21 - 40)^2 + (28.571 - 50)^2 ) = sqrt((-19)^2 + (-21.429)^2) ≈ sqrt(361 + 459.1) ≈ sqrt(820.1) ≈ 28.63

Total ≈ 29 + 41.43 + 28.63 ≈ 99.06, which is slightly higher than 98.99. So seems like the critical point is indeed a minimum.

Another perturbation: x=19, then y=600/19 ≈ 31.578. Compute:

First term: sqrt(19² + 400) = sqrt(361 + 400) = sqrt(761) ≈ 27.59

Second term: sqrt( (31.578)^2 + 900 ) ≈ sqrt(997 + 900) ≈ sqrt(1897) ≈ 43.56

Third term: sqrt( (19 - 40)^2 + (31.578 - 50)^2 ) = sqrt( (-21)^2 + (-18.422)^2 ) ≈ sqrt(441 + 339.3) ≈ sqrt(780.3) ≈ 27.93

Total ≈ 27.59 + 43.56 + 27.93 ≈ 99.08, which is again slightly higher than 98.99. So seems that (20, 30) is indeed the minimum.

But just to confirm, let me check another point. For example, x=20, y=30. Suppose I take x=20, y=30. Let me try another nearby point where xy=600, say x=25, y=24. Since 25*24=600.

Compute:

First term: sqrt(25² + 400) = sqrt(625 + 400) = sqrt(1025) ≈ 32.016

Second term: sqrt(24² + 900) = sqrt(576 + 900) = sqrt(1476) ≈ 38.42

Third term: sqrt( (25 - 40)^2 + (24 - 50)^2 ) = sqrt( (-15)^2 + (-26)^2 ) = sqrt(225 + 676) = sqrt(901) ≈ 30.02

Total ≈ 32.016 + 38.42 + 30.02 ≈ 100.46, which is higher.

So, based on these tests, it seems that the critical point at (20, 30) gives a lower value than nearby points. Hence, likely the minimum.

Now, let me check if there are other critical points or if this is the only one. Earlier, when solving the quadratic, we only got x=20 as the valid solution. So this is the only critical point in the domain.

Therefore, the minimum value is 70√2. Let me verify this:

At (20, 30):

First term: sqrt(20² + 20²) = sqrt(800) = 20√2

Second term: sqrt(30² + 30²) = sqrt(1800) = 30√2

Third term: sqrt( (-20)^2 + (-20)^2 ) = sqrt(800) = 20√2

Total: 20√2 + 30√2 + 20√2 = 70√2

Yes. So the minimum value is 70√2.

But just to be thorough, let me check if this is indeed the minimal value. Suppose there's a lower value on the boundary. However, we checked the corners and some edges, and they were all higher. Also, since our critical point is inside the domain and the function is smooth there, and the boundaries give higher values, this should be the minimum.

Hence, the minimal value is 70√2.

**Final Answer**
The minimum value is \boxed{70\sqrt{2}}.

To minimize the expression:

$$
f(x, y) = \sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{x^2 + y^2 - 80x - 100y + 4100}
$$

consider that all three terms are geometric distances. We begin by simplifying the third term’s expression:

$$
x^2 + y^2 - 80x - 100y + 4100
$$

Complete the square for the x and y terms:

- For $x$: $x^2 - 80x = (x - 40)^2 - 1600$
- For $y$: $y^2 - 100y = (y - 50)^2 - 2500$

Substituting back:

$$
x^2 + y^2 - 80x - 100y + 4100 = (x - 40)^2 + (y - 50)^2
$$

So, the third term simplifies to:

$$
\sqrt{(x - 40)^2 + (y - 50)^2}
$$

Therefore, the full expression becomes:

$$
f(x, y) = \sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{(x - 40)^2 + (y - 50)^2}
$$

---

We now interpret this expression geometrically. The form suggests that:

- $\sqrt{x^2 + 400} = \sqrt{x^2 + 20^2}$ is the Euclidean distance from $(x, 0)$ to $(0, 20)$,
- $\sqrt{y^2 + 900} = \sqrt{y^2 + 30^2}$ is the Euclidean distance from $(0, y)$ to $(0, 30)$,
- $\sqrt{(x - 40)^2 + (y - 50)^2}$ is the Euclidean distance from $(x, y)$ to $(40, 50)$.

Thus, the function can be interpreted as the sum of the distances from three different points: from $(x, 0)$ to $(0, 20)$, from $(0, y)$ to $(0, 30)$, and from $(x, y)$ to $(40, 50)$. Since $x$ and $y$ are independent variables that must be minimized, the expression becomes a three-point distance sum with variables at the endpoints.

---

### Step 1: Find Critical Point via Optimization

We proceed with partial derivatives and solve the system:

$$
\frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + 400}} + \frac{x - 40}{\sqrt{(x - 40)^2 + (y - 50)^2}} = 0
$$
$$
\frac{\partial f}{\partial y} = \frac{y}{\sqrt{y^2 + 900}} + \frac{y - 50}{\sqrt{(x - 40)^2 + (y - 50)^2}} = 0
$$

Assume the third term's square root (denote it $D$) is the same in both expressions. Then from the two equations:

$$
\frac{x}{\sqrt{x^2 + 400}} = -\frac{x - 40}{D}, \quad \frac{y}{\sqrt{y^2 + 900}} = -\frac{y - 50}{D}
$$

Squaring both equations and equating the expressions for $D^2$ gives:

$$
\frac{x^2}{x^2 + 400} = \frac{(x - 40)^2}{D^2}, \quad \frac{y^2}{y^2 + 900} = \frac{(y - 50)^2}{D^2}
$$

Equating the two right-hand sides gives:

$$
\frac{(x - 40)^2}{x^2 + 400} = \frac{(y - 50)^2}{y^2 + 900}
$$

This leads to the relation:

$$
xy = 600
$$

---

### Step 2: Solve with Constraint $xy = 600$

Now we consider this constraint $xy = 600$ and substitute into the original function.

Try specific values that satisfy this constraint and fall within the given domain $0 \leq x \leq 40$, $0 \leq y \leq 50$. We find:

- $(x, y) = (20, 30)$ satisfies $xy = 600$,
- Both values are within the allowed domain.

Now evaluate $f(20, 30)$:

- $\sqrt{20^2 + 400} = \sqrt{800} = 20\sqrt{2}$
- $\sqrt{30^2 + 900} = \sqrt{1800} = 30\sqrt{2}$
- $\sqrt{(20 - 40)^2 + (30 - 50)^2} = \sqrt{400 + 400} = \sqrt{800} = 20\sqrt{2}$

Adding these:

$$
f(20, 30) = 20\sqrt{2} + 30\sqrt{2} + 20\sqrt{2} = 70\sqrt{2}
$$

---

### Step 3: Confirm Minimality

Since the function is smooth and the domain is compact, we check boundaries and nearby points (e.g., $x = 21, y \approx 28.57$, etc.), and find that the value of $f(x, y)$ is always greater than $70\sqrt{2}$. Therefore, $(20, 30)$ is the point yielding the minimum value.

---

### Final Answer

$$
\boxed{70\sqrt{2}}
$$

[Assistant]
Explanation:
Okay, let me try to figure out how to solve this problem. The question is asking for the minimum value of the expression:

$$
\sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{x^2 + y^2 - 80x - 100y + 4100}
$$

given that $0 \leq x \leq 40$ and $0 \leq y \leq 50$.

First, I need to understand what each term represents. The first two terms look like distances. For example, $\sqrt{x^2 + 400}$ can be thought of as the distance from the point $(x, 0)$ to the point $(0, 20)$ since $\sqrt{(x - 0)^2 + (0 - 20)^2} = \sqrt{x^2 + 400}$. Similarly, $\sqrt{y^2 + 900}$ is the distance from $(0, y)$ to $(0, 30)$ because $\sqrt{(0 - 0)^2 + (y - 30)^2} = \sqrt{y^2 + 900}$. Wait, actually, if we consider coordinates, maybe we need to think differently. Let me check:

If we have a general distance formula between two points $(a, b)$ and $(c, d)$, it's $\sqrt{(a - c)^2 + (b - d)^2}$. So for the first term, if I set $x$ as the x-coordinate and the other term as the y-coordinate, but there's no y in the first term. Maybe I need to think of it as distances from some fixed points.

Alternatively, perhaps all three terms can be represented as distances in some coordinate system. Let me think.

Let me write the third term:

$$
\sqrt{x^2 + y^2 - 80x - 100y + 4100}
$$

Let me simplify the expression inside the square root:

$x^2 + y^2 - 80x - 100y + 4100$

I can complete the squares for x and y terms.

For the x terms: $x^2 - 80x$. Completing the square:

$x^2 - 80x = (x - 40)^2 - 1600$

Similarly for y terms: $y^2 - 100y = (y - 50)^2 - 2500$

So substituting back into the expression:

$$
(x - 40)^2 - 1600 + (y - 50)^2 - 2500 + 4100
$$

Calculating constants: -1600 -2500 + 4100 = (-4100) + 4100 = 0.

Therefore, the third term simplifies to:

$$
\sqrt{(x - 40)^2 + (y - 50)^2}
$$

Oh! That's interesting. So the third term is the distance between the point $(x, y)$ and the point $(40, 50)$.

So now, the entire expression is the sum of three distances:

1. $\sqrt{x^2 + 400}$: distance from $(x, 0)$ to $(0, 20)$? Wait, no. Wait, if we think of it as a distance between two points, but the first term is $\sqrt{x^2 + 400}$. Let me check again. Let me imagine that we are dealing with coordinates.

Suppose we have three points:

Let me think of this as a problem in geometry. Let me try to interpret each term as a distance.

First term: $\sqrt{x^2 + 20^2}$, so this is the distance from the point (x, 0) to (0, 20)? Wait, no. Wait, if you have a point (x, y), then the distance from (x, y) to (a, b) is sqrt[(x - a)^2 + (y - b)^2]. So if our first term is sqrt[x^2 + 400], that's sqrt[x^2 + 20^2], which would be the distance from (x, 0) to (0, 20)? Wait, but if we have (x, 0) and (0, 20), then the distance is sqrt[(x - 0)^2 + (0 - 20)^2] = sqrt(x^2 + 400). So yes, that's correct. Similarly, sqrt[y^2 + 900] is sqrt[y^2 + 30^2], so distance from (0, y) to (0, 30)? Wait, but (0, y) to (0, 30) would be |y - 30|, but sqrt(y^2 + 900) is not |y - 30|. Wait, maybe I need a different approach.

Alternatively, maybe all three terms are distances from points in some coordinate system. Let me try to think of the problem as minimizing the sum of three distances. Let me consider points in 2D space.

Suppose we have three points:

Let me define points A, B, and C such that:

- The first term is the distance from point P(x, y) to point A.

- The second term is the distance from point P(x, y) to point B.

- The third term is the distance from point P(x, y) to point C.

Wait, but the first two terms don't involve both x and y. Wait, first term is sqrt(x² + 400), which is sqrt((x - 0)^2 + (0 - 20)^2) if we think of a point (x, 0) and (0, 20). Similarly, sqrt(y² + 900) is sqrt((0 - 0)^2 + (y - 30)^2) if we consider (0, y) and (0, 30). But if we want to think of a single point (x, y), then sqrt(x² + 400) would be the distance from (x, y) to (0, 20) only if the y-coordinate is zero. But since in the problem, the variables x and y are independent, maybe this approach isn't directly applicable.

Alternatively, maybe the problem involves reflecting points or something like that. Wait, let me think again. Since the third term is the distance from (x, y) to (40, 50). Let me denote the third term as distance from (x, y) to D(40, 50).

Now, the first term sqrt(x² + 400) is the distance from (x, 0) to (0, 20). But if we think of it as a single point, maybe we need to consider another coordinate system. Alternatively, maybe think of it as the distance from (x, 0) to (0, 20) and the distance from (0, y) to (0, 30), but then how do they relate to the point (x, y)? Maybe the problem involves multiple points?

Alternatively, perhaps we need to model this as a path. Suppose we have a path that goes from (0, 20) to (x, 0) to (x, y) to (40, 50) or something like that? Wait, that seems complicated. Let me try to think of the total expression as sum of three distances.

Wait, if we consider three points:

Let me define:

- Point A at (0, 20)

- Point B at (0, 30)

- Point C at (40, 50)

Then the expression is:

sqrt[(x - 0)^2 + (0 - 20)^2] + sqrt[(y - 0)^2 + (0 - 30)^2] + sqrt[(x - 40)^2 + (y - 50)^2]

Wait, but that's not exactly. Wait, sqrt[x² + 400] is sqrt[(x - 0)^2 + (0 - 20)^2], so that's the distance between (x, 0) and (0, 20). Similarly, sqrt[y² + 900] is the distance between (0, y) and (0, 30). But then we have the distance between (x, y) and (40, 50).

But how do these relate? Maybe the variables x and y are coordinates of some point, and we have to consider distances from different points. However, the problem is that the first two terms have (x, 0) and (0, y), which are points on the axes, but the third term is from (x, y).

Alternatively, maybe think of this as a path: starting at (0, 20), going to (x, 0), then to (x, y), then to (40, 50)? But then that would be the sum of three distances, but the second term is from (x, 0) to (0, y)? Not sure. Maybe another way.

Alternatively, maybe there's a way to interpret all three terms as distances from the same point. Let me think again.

Alternatively, think of the entire expression as three distances that need to be minimized. Let me suppose that x and y are variables. So we need to minimize the sum of three distances: the distance from (x, 0) to (0, 20), the distance from (0, y) to (0, 30), and the distance from (x, y) to (40, 50). But how do these relate?

Alternatively, maybe the problem is about finding a point (x, y) such that the sum of three specific distances is minimized. Wait, if I consider (x, y) as a variable point, then:

- sqrt(x² + 400) is the distance from (x, y) to (0, 20) only if y is fixed? Wait, no. Wait, if (x, y) is a variable point, then sqrt(x² + 400) is not necessarily a distance to a fixed point unless we fix y. So perhaps there's a misinterpretation here. Maybe we need to model this differently.

Alternatively, maybe consider coordinates. Let me think about each term:

First term: sqrt(x² + 20²). If we consider that as the distance from (x, 0) to (0, 20), but then (x, 0) is a point on the x-axis. Similarly, sqrt(y² + 30²) is the distance from (0, y) to (0, 30), so that's a point on the y-axis. Then the third term is the distance between (x, y) and (40, 50).

But how do these terms relate? If we imagine that there is a path that goes from (0, 20) to (x, 0) to (x, y) to (40, 50), but that seems complicated. Maybe we need to consider reflections?

Alternatively, maybe there's a geometric interpretation where we can use the reflection principle to find minimal paths. For instance, in problems where you have to find minimal paths that reflect off a line, you can reflect a point across the line and find a straight line distance.

Wait, here's an idea. Let me think of the entire expression as the sum of three distances:

Let me consider points:

- Let’s define point P = (x, 0), Q = (0, y), and R = (x, y). However, that might not directly help.

Alternatively, perhaps think of the expression as the sum of three distances:

From (x, 0) to (0, 20) [sqrt(x² + 400)]

Plus the distance from (0, y) to (0, 30) [sqrt(y² + 900)]

Plus the distance from (x, y) to (40, 50) [sqrt((x - 40)^2 + (y - 50)^2)]

But how do these three terms relate? If we need to minimize this sum with respect to x and y? It seems that x and y are variables, but each term has different dependencies. However, maybe there is a way to relate them.

Alternatively, think of the problem as optimizing over x and y. Maybe take partial derivatives and set them to zero to find critical points? Although that might be complicated with square roots, but maybe manageable.

Alternatively, maybe there's a way to combine the terms into a single expression. Let me try to think geometrically again. Let me consider that the first term is distance from (x, 0) to (0, 20), the second term is distance from (0, y) to (0, 30), and the third term is distance from (x, y) to (40, 50). If we imagine that x and y are coordinates of a point (x, y), then maybe there are some relationships between these points.

Wait, perhaps if we consider that (x, 0) and (0, y) are projections of the point (x, y) onto the axes? Maybe not.

Alternatively, suppose that we have to go from (0, 20) to (x, 0) to (x, y) to (40, 50), and also have to go from (0, y) to (0, 30)? Not sure. This seems too vague.

Alternatively, maybe consider that the total expression is the sum of:

- Distance from (x, 0) to (0, 20)

- Distance from (0, y) to (0, 30)

- Distance from (x, y) to (40, 50)

But since x and y are variables, maybe the minimum occurs when these points are aligned in some way? For example, if we could have (x, 0) connected to (0, 20), (0, y) connected to (0, 30), and (x, y) connected to (40, 50), but I don't see a direct connection.

Alternatively, perhaps think of the entire expression as a combination of distances from different points. Let me think if there is a way to represent all three terms as parts of a single path.

Wait, here's a different idea. Let me define three points:

Let’s say we have a point A at (0, 20), point B at (0, 30), and point C at (40, 50). Now, suppose that we have a point P = (x, 0) and a point Q = (0, y). Then, maybe the total expression is the sum of distance from A to P, distance from B to Q, and distance from P to Q? Wait, no. Wait, distance from A to P is sqrt(x² + (0 - 20)^2) = sqrt(x² + 400), which matches the first term. Similarly, distance from B to Q is sqrt((0 - 0)^2 + (y - 30)^2) = sqrt(y² + 900), which matches the second term. Then the distance from P to Q is sqrt((x - 0)^2 + (0 - y)^2) = sqrt(x² + y²). But the third term is sqrt(x² + y² - 80x - 100y + 4100). Which we earlier simplified to sqrt((x - 40)^2 + (y - 50)^2). So that doesn't match.

Wait, but the third term is actually the distance from (x, y) to (40, 50). So if I consider another point, maybe D = (40, 50), then the third term is the distance between (x, y) and D. But how does that relate to the other terms?

Alternatively, maybe if I consider all three terms as distances in a coordinate system. Let me try to visualize.

Suppose we have a path that goes from point A (0, 20) to point P (x, 0), then to point Q (0, y), then to point D (40, 50). Then, the total distance would be:

Distance AP + PQ + QD. Where AP is sqrt(x² + 400), PQ is sqrt(x² + y²), and QD is sqrt((0 - 40)^2 + (y - 50)^2) = sqrt(1600 + (y - 50)^2). But that's different from our original expression, which has the third term as distance from (x, y) to (40, 50). So it's not quite matching.

Alternatively, maybe think of the expression as AP + BQ + PD, where PD is the distance from P to D. But then, PD would be sqrt((x - 40)^2 + (0 - 50)^2) = sqrt((x - 40)^2 + 2500), which is different from the third term.

Alternatively, perhaps we need to think of the third term as the distance between (x, y) and (40, 50), so if we consider a point (x, y), then the total expression is:

Distance from (x, 0) to (0, 20) + Distance from (0, y) to (0, 30) + Distance from (x, y) to (40, 50)

But how do these connect? If we imagine that (x, 0) and (0, y) are variable points, and (x, y) is another variable point? But this seems too vague.

Alternatively, maybe there's a way to relate x and y through some constraints? For example, if we consider that (x, 0) and (0, y) are connected to (x, y), but again, not sure.

Wait, going back, maybe the key is in the third term, which is the distance between (x, y) and (40, 50). So the entire expression is:

sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)^2 + (y - 50)^2)

But perhaps I need to see this as a sum of three distances from three different points to some variable point? But no, the first two terms are not distances from (x, y) to fixed points.

Wait, unless we interpret them as distances from (x, 0) to (0, 20) and from (0, y) to (0, 30). If we think of x and y as coordinates, maybe we need to find a point (x, y) such that...?

Alternatively, maybe think of this as a problem in optimization with two variables. Let me consider trying calculus here. Let me denote the function as:

f(x, y) = sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)² + (y - 50)²)

We need to find the minimum of f(x, y) over 0 ≤ x ≤ 40 and 0 ≤ y ≤ 50.

This seems like a problem that could be approached with partial derivatives. Let's try that.

First, compute the partial derivatives of f with respect to x and y, set them to zero, and solve the equations.

Let me denote:

f(x, y) = sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)² + (y - 50)²)

Let me compute ∂f/∂x:

The derivative of sqrt(x² + 400) with respect to x is (x)/sqrt(x² + 400).

The derivative of sqrt(y² + 900) with respect to x is 0.

The derivative of sqrt((x - 40)^2 + (y - 50)^2) with respect to x is [(x - 40)] / sqrt((x - 40)^2 + (y - 50)^2).

Therefore,

∂f/∂x = [x / sqrt(x² + 400)] + [(x - 40) / sqrt((x - 40)^2 + (y - 50)^2)].

Similarly, compute ∂f/∂y:

Derivative of sqrt(x² + 400) with respect to y is 0.

Derivative of sqrt(y² + 900) with respect to y is [y / sqrt(y² + 900)].

Derivative of sqrt((x - 40)^2 + (y - 50)^2) with respect to y is [(y - 50) / sqrt((x - 40)^2 + (y - 50)^2)].

Therefore,

∂f/∂y = [y / sqrt(y² + 900)] + [(y - 50) / sqrt((x - 40)^2 + (y - 50)^2)].

To find critical points, set ∂f/∂x = 0 and ∂f/∂y = 0.

So,

(1) x / sqrt(x² + 400) + (x - 40) / sqrt((x - 40)^2 + (y - 50)^2) = 0

(2) y / sqrt(y² + 900) + (y - 50) / sqrt((x - 40)^2 + (y - 50)^2) = 0

This seems quite complex. Maybe there's a way to relate equations (1) and (2). Let me denote:

Let me call the third term's square root as D = sqrt((x - 40)^2 + (y - 50)^2).

Then equations (1) and (2) become:

(1) x / sqrt(x² + 400) + (x - 40)/D = 0

(2) y / sqrt(y² + 900) + (y - 50)/D = 0

Let me rearrange equation (1):

(x)/sqrt(x² + 400) = -(x - 40)/D

Similarly, equation (2):

(y)/sqrt(y² + 900) = -(y - 50)/D

So, from (1):

Let me square both sides of equation (1):

[x² / (x² + 400)] = [(x - 40)^2]/D²

Similarly for equation (2):

[y² / (y² + 900)] = [(y - 50)^2]/D²

But D² is (x - 40)^2 + (y - 50)^2. Let me denote D² = (x - 40)^2 + (y - 50)^2.

So from equation (1):

x² / (x² + 400) = (x - 40)^2 / [(x - 40)^2 + (y - 50)^2]

Similarly, from equation (2):

y² / (y² + 900) = (y - 50)^2 / [(x - 40)^2 + (y - 50)^2]

Let me denote S = (x - 40)^2 + (y - 50)^2.

Then:

From equation (1): x² / (x² + 400) = (x - 40)^2 / S

From equation (2): y² / (y² + 900) = (y - 50)^2 / S

Let me solve for S from both equations and set them equal.

From equation (1):

S = [ (x - 40)^2 * (x² + 400) ] / x²

From equation (2):

S = [ (y - 50)^2 * (y² + 900) ] / y²

Therefore, setting them equal:

[ (x - 40)^2 * (x² + 400) ] / x² = [ (y - 50)^2 * (y² + 900) ] / y²

This seems very complicated. Maybe there's a relationship between x and y that we can exploit?

Alternatively, maybe assume that there is some proportionality between the terms. Let me think. If I denote the first fraction as something and second as something else. Alternatively, notice that 400 is 20² and 900 is 30², so maybe there's a connection with the other points.

Wait, let's think geometrically again. Let me consider the points mentioned:

- The first term is distance from (x, 0) to (0, 20)

- The second term is distance from (0, y) to (0, 30)

- The third term is distance from (x, y) to (40, 50)

But if I imagine that (x, 0) is a point on the x-axis, and (0, y) is a point on the y-axis, and (x, y) is some point in the plane. However, how do these relate?

Alternatively, maybe think of this problem as optimizing over x and y where x and y are coordinates of a point, and we have three different distances. But maybe there's a way to combine them into a single path.

Wait, here's an idea. Let me imagine that we need to find a point (x, y) such that the total expression is minimized. Let me think of the first term sqrt(x² + 400) as the distance from (x, y) to (0, 20) if y is fixed? No. Wait, if we consider the first term sqrt(x² + 400) as sqrt((x - 0)^2 + (y - 20)^2), but that would be if y is 0. Wait, no. If we set y = 0, then it would be sqrt(x² + 400). So maybe the first term is the distance from (x, 0) to (0, 20). Similarly, the second term sqrt(y² + 900) is the distance from (0, y) to (0, 30).

But if we think of x and y as coordinates such that the point (x, 0) is connected to (0, 20), and (0, y) is connected to (0, 30), and then the point (x, y) is connected to (40, 50). But how do these connect?

Alternatively, maybe think of the entire expression as the sum of three distances:

From (0, 20) to (x, 0) to (40, 50) and also from (0, 30) to (0, y) to (x, y)? This is too vague.

Alternatively, maybe think of it as three separate distances, each with their own constraints. Wait, perhaps there is a way to interpret this as minimizing the sum of three distances where the points are connected in a certain way. For example, imagine that we have to go from point A (0, 20) to some point on the x-axis (x, 0), then from there to a point (x, y), then from (x, y) to point D (40, 50), and also go from point B (0, 30) to some point on the y-axis (0, y), and then from there to (x, y). But this seems too convoluted.

Alternatively, perhaps there is an optimal point (x, y) that relates to reflections. For example, in problems where you have to find minimal paths that reflect off a surface, you can reflect points across the surface.

Let me try to think of this as a path: suppose we have to go from point A to point P on the x-axis, then from P to point Q, then from Q to point D, and also from point B to point Q on the y-axis?

Wait, here's another thought: If I consider that sqrt(x² + 400) is the distance from A(0,20) to P(x, 0), sqrt(y² + 900) is the distance from B(0,30) to Q(0, y), and sqrt((x - 40)^2 + (y - 50)^2) is the distance from Q(x, y) to D(40, 50). Wait, but Q(0, y) and Q(x, y) are different. Maybe not.

Alternatively, if I consider that there is a path from A(0,20) to P(x, 0) to D(40, 50), which would be sqrt(x² + 400) + sqrt((x - 40)^2 + (0 - 50)^2). But our third term is sqrt((x - 40)^2 + (y - 50)^2). So this is similar but with y.

Alternatively, maybe combine all three terms as paths from A to P to Q to D and B to Q? Not sure.

Alternatively, think of the entire expression as the sum of distances from A to P, B to Q, and Q to D, where P is (x, 0) and Q is (x, y). Wait, then that would be sqrt(x² + 400) + sqrt(y² + 900) + sqrt((x - 40)^2 + (y - 50)^2). But then Q is (x, y), so if we have to go from B(0, 30) to Q(0, y)? Wait, no. If Q is (x, y), then distance from B(0, 30) to Q(x, y) is sqrt(x² + (y - 30)^2). But in our expression, the second term is sqrt(y² + 900) = sqrt(y² + 30²), which is the distance from (0, y) to (0, 30). So that's a different point Q on the y-axis.

This is getting too tangled. Maybe I need to step back and think of specific values.

Let me check if the minimum occurs at some boundary points. Since x and y are constrained between 0 and 40, 0 and 50, respectively, maybe the minimum is achieved at one of the corners? For example, check at (0, 0), (40, 0), (0, 50), (40, 50), or maybe somewhere in the middle.

But before that, let me think if there's a geometric interpretation. Suppose the expression represents the sum of three distances:

- From (x, 0) to (0, 20),

- From (0, y) to (0, 30),

- From (x, y) to (40, 50).

If we imagine that (x, 0) and (0, y) are points on the axes, and (x, y) is a point. Maybe there's a relation between them. For example, if we think of the total expression as the sum of distances from (x, 0) to (0, 20), (0, y) to (0, 30), and (x, y) to (40, 50). But how?

Alternatively, think of it as:

Total distance = distance from A(0,20) to P(x,0) + distance from B(0,30) to Q(0,y) + distance from R(x,y) to D(40,50). If P, Q, R are related somehow? If P(x,0), Q(0,y), R(x,y), then R is the point (x,y). So maybe it's like the sum of distances from A to P, B to Q, and R to D, where P is on the x-axis, Q is on the y-axis, and R is the point (x,y). But how do these connect?

Alternatively, maybe think of the entire expression as the sum of distances from three different points to other points, but without a clear relationship.

Alternatively, maybe think about the following: if we fix x and y, then the expression is the sum of these three terms. Maybe there is a way to relate all three terms through coordinates.

Wait, going back to the third term, which simplifies to sqrt((x - 40)^2 + (y - 50)^2). Let me call this distance from (x, y) to (40, 50). Let me call this point E(40, 50). So the third term is distance from (x, y) to E.

Now, first term sqrt(x² + 400) is distance from (x, 0) to (0, 20). Let me call this point F(0, 20). Similarly, the second term sqrt(y² + 900) is distance from (0, y) to (0, 30). Let me call this point G(0, 30).

So the total expression is:

distance from F(0, 20) to P(x, 0) + distance from G(0, 30) to Q(0, y) + distance from R(x, y) to E(40, 50).

But how are P, Q, R related?

If I consider that P is (x, 0), Q is (0, y), and R is (x, y). So R is connected to P and Q. Maybe think of this as triangles or something. But I don't see the connection.

Alternatively, think of this as minimizing the sum of distances from F to P, from G to Q, and from R to E, where P is on x-axis, Q is on y-axis, and R is some point in the plane. But without a direct relation between P, Q, R, it's unclear.

Alternatively, maybe think about optimizing x and y such that the three terms are minimized. For instance, if we can find x and y such that the three terms have some sort of equilibrium.

Alternatively, going back to the calculus approach. Let me try to see if I can solve the equations:

From equations (1) and (2), we had:

x / sqrt(x² + 400) = -(x - 40)/D

and

y / sqrt(y² + 900) = -(y - 50)/D

Where D = sqrt((x - 40)^2 + (y - 50)^2)

Let me denote:

Let me define:

Let me set u = x / sqrt(x² + 400)

Let me set v = y / sqrt(y² + 900)

Then from equations:

u = -(x - 40)/D

v = -(y - 50)/D

So, multiplying both equations:

u * v = [(x - 40)(y - 50)] / D²

But D² = (x - 40)^2 + (y - 50)^2

So:

u * v = [(x - 40)(y - 50)] / [(x - 40)^2 + (y - 50)^2]

Hmm.

Also, note that u = x / sqrt(x² + 400) implies that u is the cosine of the angle between the vector (x, 0) and the vector to (0, 20)? Wait, not sure.

Alternatively, perhaps express u and v in terms of angles. For example, for u:

If I imagine a right triangle with base x and height 20, then u = x / sqrt(x² + 20²) = cos(theta), where theta is the angle between the hypotenuse and the x-axis. Similarly, v = y / sqrt(y² + 30²) = cos(phi), where phi is the angle between hypotenuse and y-axis.

But not sure if helpful.

Alternatively, note that from u = -(x - 40)/D and v = -(y - 50)/D, we can write:

Let me think of D as the hypotenuse of a right triangle with legs (x - 40) and (y - 50). So, u = - (x - 40)/D and v = - (y - 50)/D. So u and v are the x and y components of the unit vector pointing from (x, y) to (40, 50), but with a negative sign.

Wait, if we consider the vector from (x, y) to (40, 50), its components are (40 - x, 50 - y). The unit vector in that direction is [(40 - x)/D, (50 - y)/D]. So, according to our previous equations, u = - (x - 40)/D = (40 - x)/D, and similarly v = - (y - 50)/D = (50 - y)/D. Therefore, u and v are the components of the unit vector from (x, y) to (40, 50).

But u was defined as x / sqrt(x² + 400), which is the cosine of the angle made by the vector (x, 0) with the x-axis. Similarly, v is y / sqrt(y² + 900), which is the cosine of the angle made by (0, y) with the y-axis.

Therefore, we have:

(40 - x)/D = u = x / sqrt(x² + 400)

(50 - y)/D = v = y / sqrt(y² + 900)

Therefore:

(40 - x)/D = x / sqrt(x² + 400)

(50 - y)/D = y / sqrt(y² + 900)

Let me square both equations:

For the first equation:

[(40 - x)^2]/D² = x² / (x² + 400)

Similarly, second equation:

[(50 - y)^2]/D² = y² / (y² + 900)

But D² = (x - 40)^2 + (y - 50)^2 = (40 - x)^2 + (50 - y)^2

So from first equation:

[(40 - x)^2] / [(40 - x)^2 + (50 - y)^2] = x² / (x² + 400)

Similarly, second equation:

[(50 - y)^2] / [(40 - x)^2 + (50 - y)^2] = y² / (y² + 900)

Let me denote A = (40 - x)^2 and B = (50 - y)^2. Then D² = A + B.

So first equation:

A / (A + B) = x² / (x² + 400)

Second equation:

B / (A + B) = y² / (y² + 900)

Let me call these equations (1) and (2).

Let me denote the left-hand sides as fractions:

From equation (1): A / (A + B) = x² / (x² + 400)

From equation (2): B / (A + B) = y² / (y² + 900)

Let me add both equations:

[A / (A + B)] + [B / (A + B)] = [x² / (x² + 400)] + [y² / (y² + 900)]

Which simplifies to:

1 = [x² / (x² + 400)] + [y² / (y² + 900)]

So, this gives us an equation relating x and y:

x² / (x² + 400) + y² / (y² + 900) = 1

Let me write this as:

Let me denote u = x² and v = y². Then the equation becomes:

u / (u + 400) + v / (v + 900) = 1

Solving for u and v.

Let me solve this equation:

u / (u + 400) + v / (v + 900) = 1

Let me rearrange:

u / (u + 400) = 1 - v / (v + 900)

Compute the right side:

1 - v / (v + 900) = [(v + 900) - v] / (v + 900) = 900 / (v + 900)

Therefore:

u / (u + 400) = 900 / (v + 900)

Cross-multiplying:

u(v + 900) = 900(u + 400)

Expanding:

uv + 900u = 900u + 360000

Subtract 900u from both sides:

uv = 360000

Thus, uv = 360000

So, x² * y² = 360000

Therefore, xy = ±600. But since x and y are non-negative (between 0 and 40 for x, 0 and 50 for y), xy = 600.

So, we have that xy = 600.

Now, this is an important relation: x * y = 600.

So, we have that x * y = 600.

Now, remember that we also have from equations (1) and (2):

From equation (1): A / (A + B) = x² / (x² + 400), where A = (40 - x)^2 and B = (50 - y)^2.

But now, with xy = 600, perhaps we can find relations between x and y.

So, let me try to relate x and y using xy = 600. Since x is between 0 and 40, and y is between 0 and 50, let me check if there are possible solutions.

First, x must satisfy 0 ≤ x ≤ 40, so y = 600 / x must satisfy 0 ≤ y ≤ 50.

So, y = 600/x ≤ 50 => 600/x ≤ 50 => x ≥ 600 / 50 = 12.

Also, since x ≥ 0, and y ≥ 0, we have that x must be between 12 and 40. Because if x is between 0 and 12, y would be greater than 50, which is outside the allowed range. Therefore, x ∈ [12, 40], and y = 600 / x ∈ [12, 50] (since when x=40, y=15; when x=12, y=50). So that fits within the given constraints 0 ≤ y ≤ 50.

So, we have x ∈ [12, 40], y = 600/x.

Now, with this relation, we can substitute into one of our previous equations. Let me recall that from equation (1):

A / (A + B) = x² / (x² + 400)

But with A = (40 - x)^2 and B = (50 - y)^2 = (50 - 600/x)^2.

So, substituting:

[(40 - x)^2] / [(40 - x)^2 + (50 - 600/x)^2] = x² / (x² + 400)

This seems complicated, but maybe we can solve it.

Let me denote t = x for simplicity. Then y = 600/t.

So, equation becomes:

[(40 - t)^2] / [(40 - t)^2 + (50 - 600/t)^2] = t² / (t² + 400)

Let me cross-multiplying:

(40 - t)^2 * (t² + 400) = t² * [(40 - t)^2 + (50 - 600/t)^2]

Let me expand both sides.

First, left side:

(40 - t)^2 * (t² + 400)

Right side:

t² * [(40 - t)^2 + (50 - 600/t)^2]

Let me compute the right side first:

First compute (50 - 600/t)^2:

= (50 - 600/t)^2 = [(50t - 600)/t]^2 = (50t - 600)^2 / t²

Therefore, the right side becomes:

t² * [(40 - t)^2 + (50t - 600)^2 / t²] = t²*(40 - t)^2 + (50t - 600)^2

So, the equation is:

Left side: (40 - t)^2 * (t² + 400)

Right side: t²*(40 - t)^2 + (50t - 600)^2

Subtract right side from left side:

(40 - t)^2 * (t² + 400) - [t²*(40 - t)^2 + (50t - 600)^2] = 0

Simplify:

(40 - t)^2 * (t² + 400) - t²*(40 - t)^2 - (50t - 600)^2 = 0

Factor out (40 - t)^2:

(40 - t)^2 [t² + 400 - t²] - (50t - 600)^2 = 0

Simplify inside the brackets:

t² + 400 - t² = 400

Therefore:

(40 - t)^2 * 400 - (50t - 600)^2 = 0

Expand both terms:

First term: 400*(40 - t)^2

Second term: (50t - 600)^2

So:

400*(40 - t)^2 - (50t - 600)^2 = 0

Let me expand both terms.

First, expand 400*(40 - t)^2:

= 400*(1600 - 80t + t²) = 400*1600 - 400*80t + 400*t² = 640000 - 32000t + 400t²

Second, expand (50t - 600)^2:

= (50t)^2 - 2*50t*600 + 600² = 2500t² - 60000t + 360000

Therefore, the equation becomes:

[640000 - 32000t + 400t²] - [2500t² - 60000t + 360000] = 0

Simplify:

640000 - 32000t + 400t² - 2500t² + 60000t - 360000 = 0

Combine like terms:

t² terms: 400t² - 2500t² = -2100t²

t terms: -32000t + 60000t = 28000t

Constants: 640000 - 360000 = 280000

Thus, equation:

-2100t² + 28000t + 280000 = 0

Divide both sides by -100 to simplify:

21t² - 280t - 2800 = 0

Wait, let me check:

-2100t² / -100 = 21t²

28000t / -100 = -280t

280000 / -100 = -2800

Wait, but the original equation was:

-2100t² + 28000t + 280000 = 0

Dividing by -100:

21t² - 280t - 2800 = 0

Yes.

So we have quadratic equation:

21t² - 280t - 2800 = 0

Let me simplify this equation by dividing by common factors. Let me check GCD of coefficients 21, 280, 2800.

Factors of 21: 3, 7

Factors of 280: 2^3 * 5 * 7

Factors of 2800: 2^4 * 5^2 * 7

So common factor is 7.

Divide by 7:

3t² - 40t - 400 = 0

Now solve 3t² - 40t - 400 = 0.

Use quadratic formula: t = [40 ± sqrt(1600 + 4800)] / 6

Because discriminant D = (-40)^2 - 4*3*(-400) = 1600 + 4800 = 6400

sqrt(6400) = 80

Therefore, t = [40 ± 80] / 6

So two solutions:

t = (40 + 80)/6 = 120/6 = 20

t = (40 - 80)/6 = -40/6 = -20/3 ≈ -6.666...

But since t = x ∈ [12, 40], we discard the negative solution. Therefore, t = 20.

Thus, x = 20. Then y = 600 / x = 600 / 20 = 30.

So the critical point is at (x, y) = (20, 30).

Now we need to check if this is a minimum. Also, since our domain is 0 ≤ x ≤ 40 and 0 ≤ y ≤ 50, and we found that x must be between 12 and 40, and y between 12 and 50, this point (20, 30) lies within the domain.

Now, we need to verify if this critical point gives a minimum. Also, we should check the boundaries since the minimum could occur there.

But first, let's compute the value of the function at (20, 30):

Compute each term:

First term: sqrt(x² + 400) = sqrt(20² + 20²) = sqrt(400 + 400) = sqrt(800) = 20*sqrt(2) ≈ 28.284

Second term: sqrt(y² + 900) = sqrt(30² + 30²) = sqrt(900 + 900) = sqrt(1800) = 30*sqrt(2) ≈ 42.426

Third term: sqrt((x - 40)^2 + (y - 50)^2) = sqrt((20 - 40)^2 + (30 - 50)^2) = sqrt((-20)^2 + (-20)^2) = sqrt(400 + 400) = sqrt(800) = 20*sqrt(2) ≈ 28.284

Total sum: 20√2 + 30√2 + 20√2 = 70√2 ≈ 70 * 1.4142 ≈ 98.994

Now, let's check the boundaries. Since the minimum could be on the boundary of the domain, we need to check some corner points and edges.

First, let's check the corners:

Point (0, 0):

First term: sqrt(0 + 400) = 20

Second term: sqrt(0 + 900) = 30

Third term: sqrt((0 - 40)^2 + (0 - 50)^2) = sqrt(1600 + 2500) = sqrt(4100) ≈ 64.03

Total ≈ 20 + 30 + 64.03 ≈ 114.03

Which is greater than 70√2 ≈ 98.99.

Point (40, 50):

First term: sqrt(40² + 400) = sqrt(1600 + 400) = sqrt(2000) ≈ 44.721

Second term: sqrt(50² + 900) = sqrt(2500 + 900) = sqrt(3400) ≈ 58.309

Third term: sqrt((40 - 40)^2 + (50 - 50)^2) = 0

Total ≈ 44.721 + 58.309 + 0 ≈ 103.03, which is still higher than ≈98.99.

Another corner: (40, 0):

First term: sqrt(40² + 400) = sqrt(1600 + 400) = sqrt(2000) ≈ 44.721

Second term: sqrt(0 + 900) = 30

Third term: sqrt((40 - 40)^2 + (0 - 50)^2) = sqrt(0 + 2500) = 50

Total ≈ 44.721 + 30 + 50 ≈ 124.72, which is higher.

Similarly, (0, 50):

First term: sqrt(0 + 400) = 20

Second term: sqrt(50² + 900) = sqrt(2500 + 900) = sqrt(3400) ≈ 58.309

Third term: sqrt((0 - 40)^2 + (50 - 50)^2) = 40

Total ≈ 20 + 58.309 + 40 ≈ 118.309, which is higher.

So corners are worse. Now check edges.

First, check the edge where x = 12 (since x ≥ 12 for y ≤ 50). Wait, x ranges from 12 to 40. So maybe check if the minimal value occurs on the boundary of x=12 or y=...?

Alternatively, since we found a critical point inside the domain, and the value there is approximately 98.99, which is less than the corners, perhaps it is the minimum. But to be thorough, check some other points.

For example, check the point (20, 30): already did, value ~98.99.

Check another point on the boundary where y=50. Since y=50, then from xy=600, x=600/50=12. So (12, 50). Let's compute the value:

First term: sqrt(12² + 400) = sqrt(144 + 400) = sqrt(544) ≈ 23.32

Second term: sqrt(50² + 900) = sqrt(2500 + 900) = sqrt(3400) ≈ 58.31

Third term: sqrt((12 - 40)^2 + (50 - 50)^2) = sqrt((-28)^2 + 0) = 28

Total ≈ 23.32 + 58.31 + 28 ≈ 109.63, which is more than 98.99.

Check edge where x=40. Then y=600/40=15. So (40, 15):

First term: sqrt(40² + 400) = sqrt(1600 + 400) = sqrt(2000) ≈ 44.72

Second term: sqrt(15² + 900) = sqrt(225 + 900) = sqrt(1125) ≈ 33.54

Third term: sqrt((40 - 40)^2 + (15 - 50)^2) = sqrt(0 + (-35)^2) = 35

Total ≈ 44.72 + 33.54 + 35 ≈ 113.26, which is more than 98.99.

How about the edge where y=0. Then x=600/0 undefined, but since y cannot be 0 (since x would be infinite). But since y ≥ 0, but the minimal x for y=0 would be undefined. Hence, edge y=0 is not in our domain except at (x, 0) where x is something. Wait, maybe check point where y=0, but since from xy=600, if y=0, x is undefined. So no solution.

Similarly, for edge x=0, then y=600/0 undefined. So no solution.

Now, check another point, say, midpoint of the critical point. Wait, we already have the critical point.

Alternatively, check another point where x=20, y=30. Let me compute the value again:

First term: sqrt(20² + 20²) = sqrt(800) ≈ 28.284

Second term: sqrt(30² + 30²) = sqrt(1800) ≈ 42.426

Third term: sqrt((20-40)^2 + (30-50)^2) = sqrt(400 + 400) = sqrt(800) ≈ 28.284

Total: 28.284 + 42.426 + 28.284 ≈ 98.994, which is about 70√2 since √2 ≈ 1.4142, 70 * 1.4142 ≈ 98.994.

Now, to confirm this is indeed the minimum, let's check another point near (20, 30). Let me take x=20, y=30. Suppose we perturb a bit, say x=20 + Δx, y=30 + Δy. Let me check with Δx = 1, Δy = -1 (since xy=600, if we change x and y, need to maintain xy=600). But if we take x=21, then y=600/21 ≈ 28.571. Let's compute the value:

First term: sqrt(21² + 400) = sqrt(441 + 400) = sqrt(841) = 29

Second term: sqrt( (600/21)^2 + 900 ). Let me compute y ≈ 28.571, so y² ≈ (600/21)^2 ≈ (28.571)^2 ≈ 816.326. Then sqrt(816.326 + 900) ≈ sqrt(1716.326) ≈ 41.43

Third term: sqrt( (21 - 40)^2 + (28.571 - 50)^2 ) = sqrt((-19)^2 + (-21.429)^2) ≈ sqrt(361 + 459.1) ≈ sqrt(820.1) ≈ 28.63

Total ≈ 29 + 41.43 + 28.63 ≈ 99.06, which is slightly higher than 98.99. So seems like the critical point is indeed a minimum.

Another perturbation: x=19, then y=600/19 ≈ 31.578. Compute:

First term: sqrt(19² + 400) = sqrt(361 + 400) = sqrt(761) ≈ 27.59

Second term: sqrt( (31.578)^2 + 900 ) ≈ sqrt(997 + 900) ≈ sqrt(1897) ≈ 43.56

Third term: sqrt( (19 - 40)^2 + (31.578 - 50)^2 ) = sqrt( (-21)^2 + (-18.422)^2 ) ≈ sqrt(441 + 339.3) ≈ sqrt(780.3) ≈ 27.93

Total ≈ 27.59 + 43.56 + 27.93 ≈ 99.08, which is again slightly higher than 98.99. So seems that (20, 30) is indeed the minimum.

But just to confirm, let me check another point. For example, x=20, y=30. Suppose I take x=20, y=30. Let me try another nearby point where xy=600, say x=25, y=24. Since 25*24=600.

Compute:

First term: sqrt(25² + 400) = sqrt(625 + 400) = sqrt(1025) ≈ 32.016

Second term: sqrt(24² + 900) = sqrt(576 + 900) = sqrt(1476) ≈ 38.42

Third term: sqrt( (25 - 40)^2 + (24 - 50)^2 ) = sqrt( (-15)^2 + (-26)^2 ) = sqrt(225 + 676) = sqrt(901) ≈ 30.02

Total ≈ 32.016 + 38.42 + 30.02 ≈ 100.46, which is higher.

So, based on these tests, it seems that the critical point at (20, 30) gives a lower value than nearby points. Hence, likely the minimum.

Now, let me check if there are other critical points or if this is the only one. Earlier, when solving the quadratic, we only got x=20 as the valid solution. So this is the only critical point in the domain.

Therefore, the minimum value is 70√2. Let me verify this:

At (20, 30):

First term: sqrt(20² + 20²) = sqrt(800) = 20√2

Second term: sqrt(30² + 30²) = sqrt(1800) = 30√2

Third term: sqrt( (-20)^2 + (-20)^2 ) = sqrt(800) = 20√2

Total: 20√2 + 30√2 + 20√2 = 70√2

Yes. So the minimum value is 70√2.

But just to be thorough, let me check if this is indeed the minimal value. Suppose there's a lower value on the boundary. However, we checked the corners and some edges, and they were all higher. Also, since our critical point is inside the domain and the function is smooth there, and the boundaries give higher values, this should be the minimum.

Hence, the minimal value is 70√2.

**Final Answer**
The minimum value is \boxed{70\sqrt{2}}.

To minimize the expression:

$$
f(x, y) = \sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{x^2 + y^2 - 80x - 100y + 4100}
$$

consider that all three terms are geometric distances. We begin by simplifying the third term’s expression:

$$
x^2 + y^2 - 80x - 100y + 4100
$$

Complete the square for the x and y terms:

- For $x$: $x^2 - 80x = (x - 40)^2 - 1600$
- For $y$: $y^2 - 100y = (y - 50)^2 - 2500$

Substituting back:

$$
x^2 + y^2 - 80x - 100y + 4100 = (x - 40)^2 + (y - 50)^2
$$

So, the third term simplifies to:

$$
\sqrt{(x - 40)^2 + (y - 50)^2}
$$

Therefore, the full expression becomes:

$$
f(x, y) = \sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{(x - 40)^2 + (y - 50)^2}
$$

---

We now interpret this expression geometrically. The form suggests that:

- $\sqrt{x^2 + 400} = \sqrt{x^2 + 20^2}$ is the Euclidean distance from $(x, 0)$ to $(0, 20)$,
- $\sqrt{y^2 + 900} = \sqrt{y^2 + 30^2}$ is the Euclidean distance from $(0, y)$ to $(0, 30)$,
- $\sqrt{(x - 40)^2 + (y - 50)^2}$ is the Euclidean distance from $(x, y)$ to $(40, 50)$.

Thus, the function can be interpreted as the sum of the distances from three different points: from $(x, 0)$ to $(0, 20)$, from $(0, y)$ to $(0, 30)$, and from $(x, y)$ to $(40, 50)$. Since $x$ and $y$ are independent variables that must be minimized, the expression becomes a three-point distance sum with variables at the endpoints.

---

### Step 1: Find Critical Point via Optimization

We proceed with partial derivatives and solve the system:

$$
\frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + 400}} + \frac{x - 40}{\sqrt{(x - 40)^2 + (y - 50)^2}} = 0
$$
$$
\frac{\partial f}{\partial y} = \frac{y}{\sqrt{y^2 + 900}} + \frac{y - 50}{\sqrt{(x - 40)^2 + (y - 50)^2}} = 0
$$

Assume the third term's square root (denote it $D$) is the same in both expressions. Then from the two equations:

$$
\frac{x}{\sqrt{x^2 + 400}} = -\frac{x - 40}{D}, \quad \frac{y}{\sqrt{y^2 + 900}} = -\frac{y - 50}{D}
$$

Squaring both equations and equating the expressions for $D^2$ gives:

$$
\frac{x^2}{x^2 + 400} = \frac{(x - 40)^2}{D^2}, \quad \frac{y^2}{y^2 + 900} = \frac{(y - 50)^2}{D^2}
$$

Equating the two right-hand sides gives:

$$
\frac{(x - 40)^2}{x^2 + 400} = \frac{(y - 50)^2}{y^2 + 900}
$$

This leads to the relation:

$$
xy = 600
$$

---

### Step 2: Solve with Constraint $xy = 600$

Now we consider this constraint $xy = 600$ and substitute into the original function.

Try specific values that satisfy this constraint and fall within the given domain $0 \leq x \leq 40$, $0 \leq y \leq 50$. We find:

- $(x, y) = (20, 30)$ satisfies $xy = 600$,
- Both values are within the allowed domain.

Now evaluate $f(20, 30)$:

- $\sqrt{20^2 + 400} = \sqrt{800} = 20\sqrt{2}$
- $\sqrt{30^2 + 900} = \sqrt{1800} = 30\sqrt{2}$
- $\sqrt{(20 - 40)^2 + (30 - 50)^2} = \sqrt{400 + 400} = \sqrt{800} = 20\sqrt{2}$

Adding these:

$$
f(20, 30) = 20\sqrt{2} + 30\sqrt{2} + 20\sqrt{2} = 70\sqrt{2}
$$

---

### Step 3: Confirm Minimality

Since the function is smooth and the domain is compact, we check boundaries and nearby points (e.g., $x = 21, y \approx 28.57$, etc.), and find that the value of $f(x, y)$ is always greater than $70\sqrt{2}$. Therefore, $(20, 30)$ is the point yielding the minimum value.

---

### Final Answer

$$
\boxed{70\sqrt{2}}
$$

Solution:
Completing the square in $x$ and $y,$ the expression becomes
\[\sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{(x - 40)^2 + (y - 50)^2} = \sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{(40 - x)^2 + (50 - y)^2}.\]By QM-AM,
\begin{align*}
\sqrt{\frac{x^2 + 400}{2}} &\ge \frac{x + 20}{2}, \\
\sqrt{\frac{y^2 + 900}{2}} &\ge \frac{y + 30}{2}, \\
\sqrt{\frac{(40 - x)^2 + (50 - y)^2}{2}} &\ge \frac{(40 - x) + (50 - y)}{2},
\end{align*}so
\begin{align*}
&\sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{(40 - x)^2 + (50 - y)^2} \\
&\ge \sqrt{2} \cdot \frac{x + 20}{2} + \sqrt{2} \cdot \frac{y + 30}{2} + \sqrt{2} \cdot \frac{(40 - x) + (50 - y)}{2} \\
&= 70 \sqrt{2}.
\end{align*}Equality occurs when $x = 20$ and $y = 30,$ so the minimum value is $\boxed{70 \sqrt{2}}.$