Explanation:
Okay, so I need to find the minimum value of the function f(x) = |x - p| + |x - 15| + |x - p - 15| where 0 < p < 15, and x is between p and 15. Hmm, let me think about how to approach this.

First, I remember that functions involving absolute values can often be minimized by considering the points where the expressions inside the absolute values change their sign, which are called critical points. These critical points are usually the points where the argument of the absolute value is zero. So, for each absolute value term, I can find where it changes its behavior.

Let me write down the three absolute value terms:

1. |x - p|
2. |x - 15|
3. |x - p - 15|

So, the critical points for each term are:

1. x = p
2. x = 15
3. x = p + 15

But wait, since 0 < p < 15, then p + 15 is definitely greater than 15. So, the critical points are at x = p, x = 15, and x = p + 15. However, our domain for x is between p and 15. So, the critical points within this interval are x = p and x = 15. The third critical point, x = p + 15, is outside of our interval of interest. Therefore, within the interval [p, 15], the function f(x) is composed of different linear pieces depending on whether x is to the left or right of p and 15. But since x is between p and 15, and p is less than 15, the only critical point inside the interval is x = p. Wait, but x = 15 is also an endpoint. So maybe I need to consider the behavior of f(x) on the interval [p, 15].

Alternatively, maybe I can analyze the function f(x) by breaking it into different cases based on the value of x relative to p and 15. Let me try that.

First, since x is between p and 15, let's consider the expressions inside the absolute values:

For |x - p|: Since x ≥ p, this becomes x - p.

For |x - 15|: Since x ≤ 15, this becomes 15 - x.

For |x - p - 15|: Let's see, x - p - 15. Since x ≤ 15 and p > 0, then x - p - 15 ≤ 15 - p - 15 = -p < 0. Therefore, this absolute value becomes -(x - p - 15) = -x + p + 15.

Therefore, substituting these into f(x):

f(x) = (x - p) + (15 - x) + (-x + p + 15)

Let me simplify this:

First term: x - p

Second term: 15 - x

Third term: -x + p + 15

Adding them together:

(x - p) + (15 - x) + (-x + p + 15)

Let me compute term by term:

x - p + 15 - x - x + p + 15

Combine like terms:

x - x - x = -x

-p + p = 0

15 + 15 = 30

So overall, f(x) = -x + 30

Wait, so f(x) simplifies to -x + 30 on the interval [p, 15]?

But that seems strange. Let me check my calculations again.

Original expression:

|x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], as we said:

|x - p| = x - p

|x - 15| = 15 - x

|x - p - 15| = |x - (p + 15)|. Since x ≤ 15 and p + 15 > 15, so x - (p + 15) is negative, so absolute value is (p + 15) - x.

Therefore, the third term is (p + 15) - x.

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

Let me compute:

First term: x - p

Second term: 15 - x

Third term: p + 15 - x

Adding them:

x - p + 15 - x + p + 15 - x

Combine like terms:

x - x - x = -x

-p + p = 0

15 + 15 = 30

So indeed, f(x) = -x + 30 on [p, 15].

Wait, so if that's the case, then f(x) is a linear function decreasing from x = p to x = 15. Therefore, its minimum would be at the right endpoint, x = 15, since it's decreasing.

But let me check with specific numbers. Let me take p = 5, for example. Then f(x) = |x - 5| + |x - 15| + |x - 20|. For x between 5 and 15, |x - 5| = x - 5, |x - 15| = 15 - x, |x - 20| = 20 - x. So adding them up: (x - 5) + (15 - x) + (20 - x) = x -5 +15 -x +20 -x = (-x) + 30. So yes, same result. So f(x) = -x + 30. Therefore, on [5,15], f(x) is linear decreasing, so minimum at x=15, which is -15 + 30 = 15.

But wait, if I take x = 15, then f(15) = |15 - p| + |15 - 15| + |15 - p -15| = |15 - p| + 0 + | -p | = (15 - p) + p = 15. So that checks out. Similarly, at x = p, f(p) = |p - p| + |p - 15| + |p - p -15| = 0 + (15 - p) + | -15 | = (15 - p) + 15 = 30 - p. Which is also consistent with f(p) = -p + 30.

So if f(x) is linear decreasing on [p,15], then the minimum is at x=15, which is 15. But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So according to this, the minimum is 15? But that seems too straightforward. Maybe I made a mistake here?

Wait, but let me think again. The problem says 0 < p < 15. So p is between 0 and 15. But if f(x) is linear decreasing on [p,15], then the minimum is at x=15, which is 15. However, maybe there's a mistake in my analysis? Because sometimes with absolute value functions, the minimum can occur at a critical point inside the interval. But in this case, according to my calculation, the function simplifies to -x + 30 on the entire interval [p,15], which is linear. Therefore, the minimum is at x=15.

But let me check with another example. Let me take p = 10. Then f(x) = |x - 10| + |x - 15| + |x - 25|. For x between 10 and 15, |x - 10| = x - 10, |x - 15| = 15 - x, |x - 25| = 25 - x. Adding them up: (x - 10) + (15 - x) + (25 - x) = x -10 +15 -x +25 -x = (-x) + 30. So again, f(x) = -x + 30. So at x=15, f(x)= -15 +30=15. At x=10, f(x)= -10 +30=20. So again, decreasing. So the minimum is 15.

But wait, maybe there's a different way to think about this function? Let me try to think of f(x) as the sum of three absolute values.

Alternatively, maybe I can think of f(x) as |x - p| + |x - 15| + |x - (p + 15)|. So, three absolute values: distances from x to p, to 15, and to p + 15.

But since p + 15 is outside the interval [p,15], maybe there's a different way to interpret this.

But according to the previous analysis, on the interval [p,15], the function simplifies to -x + 30, which is linear decreasing. Therefore, the minimum is at x=15.

But let me check with p=1, for example. Let me compute f(x) at x=15: |15 -1| + |15 -15| + |15 -1 -15| = 14 + 0 + |-1| = 14 + 0 +1=15. At x= p=1, f(1)= |1 -1| + |1 -15| + |1 -1 -15| = 0 +14 + |-15| = 0 +14 +15=29. So again, decreasing from 29 to 15.

But maybe there's a mistake here? Because if I think of the function as sum of absolute values, usually the minimum occurs at the median of the points. But here, the points are p, 15, and p + 15. So the three points are p, 15, p + 15. The median of these three points would be 15, since p < 15 < p + 15. Therefore, the minimum of the sum of absolute deviations occurs at the median, which is 15. Therefore, the minimum value is f(15) = 15.

But wait, that seems to confirm the previous result. Therefore, maybe the answer is 15? But the problem says "for x in the interval p ≤ x ≤15". So if the minimum occurs at x=15, which is in the interval, then the minimum value is 15.

But let me check again. Suppose p is 7.5, so halfway between 0 and 15. Then f(x) = |x - 7.5| + |x - 15| + |x - 22.5|. For x between 7.5 and 15, the function is -x + 30. So at x=15, f(x)= -15 +30=15. If I compute f(15) directly, it's |15 -7.5| + |15 -15| + |15 -22.5| = 7.5 + 0 + 7.5 = 15. Correct.

But let me think again. Suppose I take x somewhere in between. Let me take x=10, p=7.5. Then f(10)= |10 -7.5| + |10 -15| + |10 -22.5| = 2.5 + 5 + 12.5 = 20. Which is equal to -10 +30=20. Correct. So yes, the function is linear decreasing.

But then why does the problem specify 0 < p <15? Maybe there's a case where the minimum is not at x=15? Wait, but according to the analysis, regardless of p, as long as x is between p and 15, the function simplifies to -x +30, which is decreasing. Therefore, the minimum is at x=15.

But maybe I made a mistake in simplifying the function? Let me check again.

Original function: |x - p| + |x - 15| + |x - p -15|.

For x in [p,15], |x - p| = x - p, |x -15| = 15 -x, and |x - p -15| = |x - (p +15)|. Since x ≤15 and p +15 >15, then x - (p +15) is negative, so absolute value is (p +15) -x. Therefore, adding them up:

(x - p) + (15 -x) + (p +15 -x) = x - p +15 -x + p +15 -x = (-x) + 30.

Yes, that's correct. Therefore, f(x) = -x +30 on [p,15]. Therefore, the function is linear decreasing, so minimum at x=15, which is 15.

But then why does the problem mention 0 < p <15? Maybe there's a different interpretation? Or maybe there's a mistake in my analysis?

Wait, let me think again. Suppose p is very close to 0, say p approaches 0. Then the function f(x) = |x - p| + |x -15| + |x - p -15|. For x in [p,15], which is [near 0, 15]. Then, as before, f(x) = -x +30. So the minimum is still 15.

Alternatively, maybe the problem is expecting me to consider that when x is between p and 15, but maybe there's a different way to express the function?

Alternatively, maybe I need to check if there's a point inside the interval [p,15] where the function could have a minimum? But according to the expression f(x) = -x +30, which is linear, there are no minima inside the interval except at the endpoints. Since it's decreasing, the minimum is at the right endpoint.

Alternatively, maybe I need to check if there's a mistake in assuming that |x - p -15| is equal to p +15 -x. Let me check with x=10, p=5. Then |10 -5 -15| = | -10 | =10. And p +15 -x =5 +15 -10=10. Correct.

Another example: x=15, p=5. |15 -5 -15| = |-5| =5. p +15 -x=5 +15 -15=5. Correct.

So that seems correct.

Therefore, according to all these checks, the function simplifies to -x +30 on [p,15], which is linear decreasing, so minimum at x=15, value 15.

But the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So maybe the answer is 15? But let me check again with another approach.

Alternatively, maybe I can think of f(x) as the sum of three absolute values. The sum of absolute values is minimized at the median of the points. The three points are p, 15, and p +15. The median of these three numbers is 15, since p <15 <p +15. Therefore, the minimum of f(x) occurs at x=15, and the minimum value is f(15)= |15 -p| + |15 -15| + |15 -p -15| = (15 -p) + 0 + | -p | = 15 -p + p =15.

Yes, that's another way to see it. Therefore, regardless of p, the minimum is 15.

But then why does the problem specify 0 < p <15? Maybe to ensure that p +15 is not overlapping with the interval? Wait, if p were 15, then p +15 would be 30, but since p <15, p +15 is between 15 and 30. So the three points are p, 15, and p +15, with p <15 <p +15. Therefore, the median is 15, so the minimum occurs at 15.

Therefore, the answer should be 15.

But let me check once more with p=1, x=15. f(15)= |15 -1| + |15 -15| + |15 -1 -15| =14 +0 + |-1|=14 +0 +1=15. Correct.

Alternatively, if I take p=14, which is close to 15. Then f(x) = |x -14| + |x -15| + |x -29|. For x in [14,15], |x -14| =x -14, |x -15|=15 -x, |x -29|=29 -x. Adding them up: (x -14) + (15 -x) + (29 -x) = x -14 +15 -x +29 -x = (-x) + 30. So again, f(x) = -x +30. At x=15, f(x)= -15 +30=15. At x=14, f(x)= -14 +30=16. So again, minimum at x=15.

Therefore, regardless of p, the minimum is 15.

But maybe the problem is more complicated? Maybe I need to check if there's a case where the minimum occurs inside the interval? For example, if the function wasn't linear? But according to the previous analysis, it is linear.

Wait, but let me think again. Suppose that the function f(x) is not linear? For example, if there was a critical point inside the interval [p,15], then the function could have a minimum there. But according to the previous analysis, the function simplifies to -x +30, which is linear. Therefore, there are no critical points inside the interval.

But why is that? Because when we simplified the absolute values, we assumed that x is between p and 15, so the expressions inside the absolute values have consistent signs. Therefore, the function is linear.

Alternatively, maybe there's a mistake in assuming that |x - p -15| is equal to p +15 -x? Let me check with x= p. If x=p, then |p - p -15| = |-15| =15. And p +15 -x = p +15 -p =15. Correct. If x=15, then |15 -p -15| = |-p| =p. And p +15 -x = p +15 -15 =p. Correct. So that seems correct.

Therefore, I think my analysis is correct. Therefore, the minimum value is 15.

But let me check with calculus. If I consider f(x) = -x +30 on [p,15], then derivative is -1, which is negative, so function is decreasing. Therefore, minimum at x=15.

Alternatively, if I didn't simplify the function, but tried to take derivative, but since absolute value functions are not differentiable at critical points, but in the interval [p,15], the function is differentiable, and derivative is -1.

Therefore, conclusion: the minimum value is 15.

But the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So maybe the answer is 15? But I need to make sure.

Alternatively, maybe there's a mistake in the problem statement? Or maybe I misread it? Let me check again.

The function is f(x)=|x - p| + |x -15| + |x - p -15|. The interval is p ≤x ≤15.

Yes.

Alternatively, maybe the problem is expecting me to consider that when x is between p and 15, the third term |x - p -15| can be written as |(x - p) -15|. But since x - p is between 0 and 15 - p, so (x - p) -15 is between -15 and -p. Therefore, absolute value is 15 - (x - p) = 15 -x + p. Which is same as before.

Therefore, adding up:

|x - p| + |x -15| + |x - p -15| = (x - p) + (15 -x) + (15 -x + p) = x - p +15 -x +15 -x +p = (-x) + 30.

Yes.

Therefore, I think my conclusion is correct. The minimum value is 15.

But maybe the answer is different? Let me check with p=0. Wait, p is greater than 0. If p approaches 0, then the minimum is still 15.

Alternatively, maybe there's a different approach. Let me think of f(x) as the sum of three absolute values.

Let me denote the three points as a = p, b =15, c = p +15. Then f(x) = |x - a| + |x - b| + |x - c|.

As mentioned before, the sum of absolute deviations is minimized at the median of the points. Since a < b < c, the median is b =15. Therefore, the minimum occurs at x=15, and the minimum value is f(15) = |15 -a| + |15 -b| + |15 -c| = (15 -a) + 0 + (c -15) = (15 -a) + (c -15) = c -a.

But c = p +15, a = p. Therefore, c -a = p +15 -p =15.

Therefore, the minimum value is 15.

Therefore, regardless of p, the minimum value is 15.

Therefore, the answer is 15.

But the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So maybe the answer is 15?

But let me check once again with p=1, x=15. f(15)= |15 -1| + |15 -15| + |15 -1 -15| =14 +0 +1=15. Correct.

Alternatively, if I take p=2, x=15: |15 -2| + |15 -15| + |15 -2 -15| =13 +0 + |-2|=13 +0 +2=15. Correct.

Therefore, yes, the minimum value is 15.

Therefore, I think the answer is 15.

But why does the problem specify 0 < p <15? Maybe to ensure that p +15 is not overlapping with the interval? For example, if p were 15, then p +15 would be 30, but since p <15, p +15 is between 15 and 30. Therefore, the three points are p, 15, p +15, with p <15 <p +15. Therefore, the median is 15.

Therefore, the answer is 15.

But maybe the problem is expecting me to write it as 15? Or maybe there's a trick here?

Alternatively, maybe I need to check if there's a case where the minimum occurs at x=p? For example, if the function was increasing? But according to our previous analysis, the function is decreasing on [p,15], so minimum at x=15.

Alternatively, maybe there's a mistake in the problem? Or maybe I need to check once again.

Wait, let me think of another way. Suppose I don't simplify the function. Let me consider f(x) = |x - p| + |x -15| + |x - p -15|. Let me think of this as |x - p| + |x -15| + |x - (p +15)|.

If I think of this as the sum of distances from x to three points: p, 15, and p +15. The sum of distances is minimized at the median. Since p <15 <p +15, the median is 15. Therefore, the minimum occurs at x=15, and the value is the sum of distances from 15 to each of the three points.

Distance from 15 to p is 15 - p.

Distance from 15 to 15 is 0.

Distance from 15 to p +15 is p +15 -15 = p.

Therefore, total distance is (15 - p) + 0 + p =15.

Therefore, the minimum value is 15.

Therefore, regardless of p, the minimum value is 15.

Therefore, the answer is 15.

But maybe the problem is expecting me to write \boxed{15}?

Alternatively, maybe I made a mistake in assuming that the median is the minimizer? Let me recall that for the sum of absolute deviations, the minimum occurs at the median. For an odd number of points, it's the middle one. For even number, any point between the two middle ones.

In this case, three points: p, 15, p +15. The median is 15. Therefore, the minimum occurs at 15.

Therefore, the answer is 15.

Therefore, I think the answer is 15.

But maybe the problem is more complex? Let me check once again.

Wait, suppose p=7.5, and x=15. Then f(x)= |15 -7.5| + |15 -15| + |15 -7.5 -15| =7.5 +0 + |-7.5|=7.5 +0 +7.5=15. Correct.

Alternatively, if I take x=10, p=7.5, then f(x)= |10 -7.5| + |10 -15| + |10 -7.5 -15| =2.5 +5 + |-7.5|=2.5 +5 +7.5=15. Wait, that's 15? Wait, but according to previous analysis, f(x) = -x +30. At x=10, that would be -10 +30=20. But here, I get 15? Wait, that's a contradiction!

Wait, what? Wait, if p=7.5, x=10, then:

First term: |10 -7.5| =2.5

Second term: |10 -15|=5

Third term: |10 -7.5 -15|=|10 -22.5|=| -12.5 | =12.5

Therefore, total is 2.5 +5 +12.5=20. Which is equal to -10 +30=20. So my previous calculation was wrong. I thought the third term was |-7.5|, but that's incorrect.

Wait, |x - p -15| = |10 -7.5 -15| = |10 -22.5| = |-12.5| =12.5. So total is 2.5 +5 +12.5=20. Which matches -x +30. So my previous calculation where I thought it was |-7.5| was wrong.

Therefore, that was a miscalculation. Therefore, the previous conclusion is correct.

Therefore, the answer is 15.

Therefore, I think I can be confident that the minimum value is 15.

**Final Answer**
\boxed{15}

To determine the **minimum value** of the function
$$
f(x) = |x - p| + |x - 15| + |x - p - 15|
$$
for $ x \in [p, 15] $, where $ 0 < p < 15 $, we proceed by analyzing the behavior of the function within the given interval.

---

### Step 1: Understand the Structure of the Function

The function is a sum of three absolute value terms:

- $ |x - p| $
- $ |x - 15| $
- $ |x - p - 15| $

We are to minimize this function over the interval $ x \in [p, 15] $.

---

### Step 2: Analyze the Behavior of Each Term in the Interval

Let’s evaluate the sign of each expression inside the absolute values for $ x \in [p, 15] $:

- $ x - p \geq 0 $, since $ x \geq p $, so $ |x - p| = x - p $
- $ x - 15 \leq 0 $, since $ x \leq 15 $, so $ |x - 15| = 15 - x $
- $ x - p - 15 \leq 0 $, since $ x \leq 15 $ and $ p > 0 $, so $ |x - p - 15| = p + 15 - x $

Substituting these into the function:

$$
f(x) = (x - p) + (15 - x) + (p + 15 - x)
$$

Simplify:

$$
f(x) = x - p + 15 - x + p + 15 - x = -x + 30
$$

So, on the interval $ [p, 15] $, the function simplifies to:

$$
f(x) = -x + 30
$$

---

### Step 3: Determine the Minimum of the Simplified Function

The function $ f(x) = -x + 30 $ is a **linear function** with a **negative slope**. Therefore, it is **strictly decreasing** on the interval $ [p, 15] $.

This means the **minimum value** occurs at the **right endpoint** of the interval, i.e., at $ x = 15 $.

---

### Step 4: Compute the Minimum Value

$$
f(15) = -15 + 30 = 15
$$

---

### Step 5: Confirm with an Alternative Approach

We can also interpret the function as the **sum of distances** from $ x $ to three points: $ p $, $ 15 $, and $ p + 15 $. The sum of absolute deviations is minimized at the **median** of the three points.

Since $ p < 15 < p + 15 $, the median is $ 15 $, and the minimum value of the function occurs at $ x = 15 $.

Evaluating the function at $ x = 15 $:

$$
f(15) = |15 - p| + |15 - 15| + |15 - p - 15| = (15 - p) + 0 + p = 15
$$

---

### Final Answer

$$
\boxed{15}
$$

[Assistant]
Explanation:
Okay, so I need to find the minimum value of the function f(x) = |x - p| + |x - 15| + |x - p - 15| where 0 < p < 15, and x is between p and 15. Hmm, let me think about how to approach this.

First, I remember that functions involving absolute values can often be minimized by considering the points where the expressions inside the absolute values change their sign, which are called critical points. These critical points are usually the points where the argument of the absolute value is zero. So, for each absolute value term, I can find where it changes its behavior.

Let me write down the three absolute value terms:

1. |x - p|
2. |x - 15|
3. |x - p - 15|

So, the critical points for each term are:

1. x = p
2. x = 15
3. x = p + 15

But wait, since 0 < p < 15, then p + 15 is definitely greater than 15. So, the critical points are at x = p, x = 15, and x = p + 15. However, our domain for x is between p and 15. So, the critical points within this interval are x = p and x = 15. The third critical point, x = p + 15, is outside of our interval of interest. Therefore, within the interval [p, 15], the function f(x) is composed of different linear pieces depending on whether x is to the left or right of p and 15. But since x is between p and 15, and p is less than 15, the only critical point inside the interval is x = p. Wait, but x = 15 is also an endpoint. So maybe I need to consider the behavior of f(x) on the interval [p, 15].

Alternatively, maybe I can analyze the function f(x) by breaking it into different cases based on the value of x relative to p and 15. Let me try that.

First, since x is between p and 15, let's consider the expressions inside the absolute values:

For |x - p|: Since x ≥ p, this becomes x - p.

For |x - 15|: Since x ≤ 15, this becomes 15 - x.

For |x - p - 15|: Let's see, x - p - 15. Since x ≤ 15 and p > 0, then x - p - 15 ≤ 15 - p - 15 = -p < 0. Therefore, this absolute value becomes -(x - p - 15) = -x + p + 15.

Therefore, substituting these into f(x):

f(x) = (x - p) + (15 - x) + (-x + p + 15)

Let me simplify this:

First term: x - p

Second term: 15 - x

Third term: -x + p + 15

Adding them together:

(x - p) + (15 - x) + (-x + p + 15)

Let me compute term by term:

x - p + 15 - x - x + p + 15

Combine like terms:

x - x - x = -x

-p + p = 0

15 + 15 = 30

So overall, f(x) = -x + 30

Wait, so f(x) simplifies to -x + 30 on the interval [p, 15]?

But that seems strange. Let me check my calculations again.

Original expression:

|x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], as we said:

|x - p| = x - p

|x - 15| = 15 - x

|x - p - 15| = |x - (p + 15)|. Since x ≤ 15 and p + 15 > 15, so x - (p + 15) is negative, so absolute value is (p + 15) - x.

Therefore, the third term is (p + 15) - x.

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

Let me compute:

First term: x - p

Second term: 15 - x

Third term: p + 15 - x

Adding them:

x - p + 15 - x + p + 15 - x

Combine like terms:

x - x - x = -x

-p + p = 0

15 + 15 = 30

So indeed, f(x) = -x + 30 on [p, 15].

Wait, so if that's the case, then f(x) is a linear function decreasing from x = p to x = 15. Therefore, its minimum would be at the right endpoint, x = 15, since it's decreasing.

But let me check with specific numbers. Let me take p = 5, for example. Then f(x) = |x - 5| + |x - 15| + |x - 20|. For x between 5 and 15, |x - 5| = x - 5, |x - 15| = 15 - x, |x - 20| = 20 - x. So adding them up: (x - 5) + (15 - x) + (20 - x) = x -5 +15 -x +20 -x = (-x) + 30. So yes, same result. So f(x) = -x + 30. Therefore, on [5,15], f(x) is linear decreasing, so minimum at x=15, which is -15 + 30 = 15.

But wait, if I take x = 15, then f(15) = |15 - p| + |15 - 15| + |15 - p -15| = |15 - p| + 0 + | -p | = (15 - p) + p = 15. So that checks out. Similarly, at x = p, f(p) = |p - p| + |p - 15| + |p - p -15| = 0 + (15 - p) + | -15 | = (15 - p) + 15 = 30 - p. Which is also consistent with f(p) = -p + 30.

So if f(x) is linear decreasing on [p,15], then the minimum is at x=15, which is 15. But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So according to this, the minimum is 15? But that seems too straightforward. Maybe I made a mistake here?

Wait, but let me think again. The problem says 0 < p < 15. So p is between 0 and 15. But if f(x) is linear decreasing on [p,15], then the minimum is at x=15, which is 15. However, maybe there's a mistake in my analysis? Because sometimes with absolute value functions, the minimum can occur at a critical point inside the interval. But in this case, according to my calculation, the function simplifies to -x + 30 on the entire interval [p,15], which is linear. Therefore, the minimum is at x=15.

But let me check with another example. Let me take p = 10. Then f(x) = |x - 10| + |x - 15| + |x - 25|. For x between 10 and 15, |x - 10| = x - 10, |x - 15| = 15 - x, |x - 25| = 25 - x. Adding them up: (x - 10) + (15 - x) + (25 - x) = x -10 +15 -x +25 -x = (-x) + 30. So again, f(x) = -x + 30. So at x=15, f(x)= -15 +30=15. At x=10, f(x)= -10 +30=20. So again, decreasing. So the minimum is 15.

But wait, maybe there's a different way to think about this function? Let me try to think of f(x) as the sum of three absolute values.

Alternatively, maybe I can think of f(x) as |x - p| + |x - 15| + |x - (p + 15)|. So, three absolute values: distances from x to p, to 15, and to p + 15.

But since p + 15 is outside the interval [p,15], maybe there's a different way to interpret this.

But according to the previous analysis, on the interval [p,15], the function simplifies to -x + 30, which is linear decreasing. Therefore, the minimum is at x=15.

But let me check with p=1, for example. Let me compute f(x) at x=15: |15 -1| + |15 -15| + |15 -1 -15| = 14 + 0 + |-1| = 14 + 0 +1=15. At x= p=1, f(1)= |1 -1| + |1 -15| + |1 -1 -15| = 0 +14 + |-15| = 0 +14 +15=29. So again, decreasing from 29 to 15.

But maybe there's a mistake here? Because if I think of the function as sum of absolute values, usually the minimum occurs at the median of the points. But here, the points are p, 15, and p + 15. So the three points are p, 15, p + 15. The median of these three points would be 15, since p < 15 < p + 15. Therefore, the minimum of the sum of absolute deviations occurs at the median, which is 15. Therefore, the minimum value is f(15) = 15.

But wait, that seems to confirm the previous result. Therefore, maybe the answer is 15? But the problem says "for x in the interval p ≤ x ≤15". So if the minimum occurs at x=15, which is in the interval, then the minimum value is 15.

But let me check again. Suppose p is 7.5, so halfway between 0 and 15. Then f(x) = |x - 7.5| + |x - 15| + |x - 22.5|. For x between 7.5 and 15, the function is -x + 30. So at x=15, f(x)= -15 +30=15. If I compute f(15) directly, it's |15 -7.5| + |15 -15| + |15 -22.5| = 7.5 + 0 + 7.5 = 15. Correct.

But let me think again. Suppose I take x somewhere in between. Let me take x=10, p=7.5. Then f(10)= |10 -7.5| + |10 -15| + |10 -22.5| = 2.5 + 5 + 12.5 = 20. Which is equal to -10 +30=20. Correct. So yes, the function is linear decreasing.

But then why does the problem specify 0 < p <15? Maybe there's a case where the minimum is not at x=15? Wait, but according to the analysis, regardless of p, as long as x is between p and 15, the function simplifies to -x +30, which is decreasing. Therefore, the minimum is at x=15.

But maybe I made a mistake in simplifying the function? Let me check again.

Original function: |x - p| + |x - 15| + |x - p -15|.

For x in [p,15], |x - p| = x - p, |x -15| = 15 -x, and |x - p -15| = |x - (p +15)|. Since x ≤15 and p +15 >15, then x - (p +15) is negative, so absolute value is (p +15) -x. Therefore, adding them up:

(x - p) + (15 -x) + (p +15 -x) = x - p +15 -x + p +15 -x = (-x) + 30.

Yes, that's correct. Therefore, f(x) = -x +30 on [p,15]. Therefore, the function is linear decreasing, so minimum at x=15, which is 15.

But then why does the problem mention 0 < p <15? Maybe there's a different interpretation? Or maybe there's a mistake in my analysis?

Wait, let me think again. Suppose p is very close to 0, say p approaches 0. Then the function f(x) = |x - p| + |x -15| + |x - p -15|. For x in [p,15], which is [near 0, 15]. Then, as before, f(x) = -x +30. So the minimum is still 15.

Alternatively, maybe the problem is expecting me to consider that when x is between p and 15, but maybe there's a different way to express the function?

Alternatively, maybe I need to check if there's a point inside the interval [p,15] where the function could have a minimum? But according to the expression f(x) = -x +30, which is linear, there are no minima inside the interval except at the endpoints. Since it's decreasing, the minimum is at the right endpoint.

Alternatively, maybe I need to check if there's a mistake in assuming that |x - p -15| is equal to p +15 -x. Let me check with x=10, p=5. Then |10 -5 -15| = | -10 | =10. And p +15 -x =5 +15 -10=10. Correct.

Another example: x=15, p=5. |15 -5 -15| = |-5| =5. p +15 -x=5 +15 -15=5. Correct.

So that seems correct.

Therefore, according to all these checks, the function simplifies to -x +30 on [p,15], which is linear decreasing, so minimum at x=15, value 15.

But the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So maybe the answer is 15? But let me check again with another approach.

Alternatively, maybe I can think of f(x) as the sum of three absolute values. The sum of absolute values is minimized at the median of the points. The three points are p, 15, and p +15. The median of these three numbers is 15, since p <15 <p +15. Therefore, the minimum of f(x) occurs at x=15, and the minimum value is f(15)= |15 -p| + |15 -15| + |15 -p -15| = (15 -p) + 0 + | -p | = 15 -p + p =15.

Yes, that's another way to see it. Therefore, regardless of p, the minimum is 15.

But then why does the problem specify 0 < p <15? Maybe to ensure that p +15 is not overlapping with the interval? Wait, if p were 15, then p +15 would be 30, but since p <15, p +15 is between 15 and 30. So the three points are p, 15, and p +15, with p <15 <p +15. Therefore, the median is 15, so the minimum occurs at 15.

Therefore, the answer should be 15.

But let me check once more with p=1, x=15. f(15)= |15 -1| + |15 -15| + |15 -1 -15| =14 +0 + |-1|=14 +0 +1=15. Correct.

Alternatively, if I take p=14, which is close to 15. Then f(x) = |x -14| + |x -15| + |x -29|. For x in [14,15], |x -14| =x -14, |x -15|=15 -x, |x -29|=29 -x. Adding them up: (x -14) + (15 -x) + (29 -x) = x -14 +15 -x +29 -x = (-x) + 30. So again, f(x) = -x +30. At x=15, f(x)= -15 +30=15. At x=14, f(x)= -14 +30=16. So again, minimum at x=15.

Therefore, regardless of p, the minimum is 15.

But maybe the problem is more complicated? Maybe I need to check if there's a case where the minimum occurs inside the interval? For example, if the function wasn't linear? But according to the previous analysis, it is linear.

Wait, but let me think again. Suppose that the function f(x) is not linear? For example, if there was a critical point inside the interval [p,15], then the function could have a minimum there. But according to the previous analysis, the function simplifies to -x +30, which is linear. Therefore, there are no critical points inside the interval.

But why is that? Because when we simplified the absolute values, we assumed that x is between p and 15, so the expressions inside the absolute values have consistent signs. Therefore, the function is linear.

Alternatively, maybe there's a mistake in assuming that |x - p -15| is equal to p +15 -x? Let me check with x= p. If x=p, then |p - p -15| = |-15| =15. And p +15 -x = p +15 -p =15. Correct. If x=15, then |15 -p -15| = |-p| =p. And p +15 -x = p +15 -15 =p. Correct. So that seems correct.

Therefore, I think my analysis is correct. Therefore, the minimum value is 15.

But let me check with calculus. If I consider f(x) = -x +30 on [p,15], then derivative is -1, which is negative, so function is decreasing. Therefore, minimum at x=15.

Alternatively, if I didn't simplify the function, but tried to take derivative, but since absolute value functions are not differentiable at critical points, but in the interval [p,15], the function is differentiable, and derivative is -1.

Therefore, conclusion: the minimum value is 15.

But the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So maybe the answer is 15? But I need to make sure.

Alternatively, maybe there's a mistake in the problem statement? Or maybe I misread it? Let me check again.

The function is f(x)=|x - p| + |x -15| + |x - p -15|. The interval is p ≤x ≤15.

Yes.

Alternatively, maybe the problem is expecting me to consider that when x is between p and 15, the third term |x - p -15| can be written as |(x - p) -15|. But since x - p is between 0 and 15 - p, so (x - p) -15 is between -15 and -p. Therefore, absolute value is 15 - (x - p) = 15 -x + p. Which is same as before.

Therefore, adding up:

|x - p| + |x -15| + |x - p -15| = (x - p) + (15 -x) + (15 -x + p) = x - p +15 -x +15 -x +p = (-x) + 30.

Yes.

Therefore, I think my conclusion is correct. The minimum value is 15.

But maybe the answer is different? Let me check with p=0. Wait, p is greater than 0. If p approaches 0, then the minimum is still 15.

Alternatively, maybe there's a different approach. Let me think of f(x) as the sum of three absolute values.

Let me denote the three points as a = p, b =15, c = p +15. Then f(x) = |x - a| + |x - b| + |x - c|.

As mentioned before, the sum of absolute deviations is minimized at the median of the points. Since a < b < c, the median is b =15. Therefore, the minimum occurs at x=15, and the minimum value is f(15) = |15 -a| + |15 -b| + |15 -c| = (15 -a) + 0 + (c -15) = (15 -a) + (c -15) = c -a.

But c = p +15, a = p. Therefore, c -a = p +15 -p =15.

Therefore, the minimum value is 15.

Therefore, regardless of p, the minimum value is 15.

Therefore, the answer is 15.

But the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤15." So maybe the answer is 15?

But let me check once again with p=1, x=15. f(15)= |15 -1| + |15 -15| + |15 -1 -15| =14 +0 +1=15. Correct.

Alternatively, if I take p=2, x=15: |15 -2| + |15 -15| + |15 -2 -15| =13 +0 + |-2|=13 +0 +2=15. Correct.

Therefore, yes, the minimum value is 15.

Therefore, I think the answer is 15.

But why does the problem specify 0 < p <15? Maybe to ensure that p +15 is not overlapping with the interval? For example, if p were 15, then p +15 would be 30, but since p <15, p +15 is between 15 and 30. Therefore, the three points are p, 15, p +15, with p <15 <p +15. Therefore, the median is 15.

Therefore, the answer is 15.

But maybe the problem is expecting me to write it as 15? Or maybe there's a trick here?

Alternatively, maybe I need to check if there's a case where the minimum occurs at x=p? For example, if the function was increasing? But according to our previous analysis, the function is decreasing on [p,15], so minimum at x=15.

Alternatively, maybe there's a mistake in the problem? Or maybe I need to check once again.

Wait, let me think of another way. Suppose I don't simplify the function. Let me consider f(x) = |x - p| + |x -15| + |x - p -15|. Let me think of this as |x - p| + |x -15| + |x - (p +15)|.

If I think of this as the sum of distances from x to three points: p, 15, and p +15. The sum of distances is minimized at the median. Since p <15 <p +15, the median is 15. Therefore, the minimum occurs at x=15, and the value is the sum of distances from 15 to each of the three points.

Distance from 15 to p is 15 - p.

Distance from 15 to 15 is 0.

Distance from 15 to p +15 is p +15 -15 = p.

Therefore, total distance is (15 - p) + 0 + p =15.

Therefore, the minimum value is 15.

Therefore, regardless of p, the minimum value is 15.

Therefore, the answer is 15.

But maybe the problem is expecting me to write \boxed{15}?

Alternatively, maybe I made a mistake in assuming that the median is the minimizer? Let me recall that for the sum of absolute deviations, the minimum occurs at the median. For an odd number of points, it's the middle one. For even number, any point between the two middle ones.

In this case, three points: p, 15, p +15. The median is 15. Therefore, the minimum occurs at 15.

Therefore, the answer is 15.

Therefore, I think the answer is 15.

But maybe the problem is more complex? Let me check once again.

Wait, suppose p=7.5, and x=15. Then f(x)= |15 -7.5| + |15 -15| + |15 -7.5 -15| =7.5 +0 + |-7.5|=7.5 +0 +7.5=15. Correct.

Alternatively, if I take x=10, p=7.5, then f(x)= |10 -7.5| + |10 -15| + |10 -7.5 -15| =2.5 +5 + |-7.5|=2.5 +5 +7.5=15. Wait, that's 15? Wait, but according to previous analysis, f(x) = -x +30. At x=10, that would be -10 +30=20. But here, I get 15? Wait, that's a contradiction!

Wait, what? Wait, if p=7.5, x=10, then:

First term: |10 -7.5| =2.5

Second term: |10 -15|=5

Third term: |10 -7.5 -15|=|10 -22.5|=| -12.5 | =12.5

Therefore, total is 2.5 +5 +12.5=20. Which is equal to -10 +30=20. So my previous calculation was wrong. I thought the third term was |-7.5|, but that's incorrect.

Wait, |x - p -15| = |10 -7.5 -15| = |10 -22.5| = |-12.5| =12.5. So total is 2.5 +5 +12.5=20. Which matches -x +30. So my previous calculation where I thought it was |-7.5| was wrong.

Therefore, that was a miscalculation. Therefore, the previous conclusion is correct.

Therefore, the answer is 15.

Therefore, I think I can be confident that the minimum value is 15.

**Final Answer**
\boxed{15}

To determine the **minimum value** of the function
$$
f(x) = |x - p| + |x - 15| + |x - p - 15|
$$
for $ x \in [p, 15] $, where $ 0 < p < 15 $, we proceed by analyzing the behavior of the function within the given interval.

---

### Step 1: Understand the Structure of the Function

The function is a sum of three absolute value terms:

- $ |x - p| $
- $ |x - 15| $
- $ |x - p - 15| $

We are to minimize this function over the interval $ x \in [p, 15] $.

---

### Step 2: Analyze the Behavior of Each Term in the Interval

Let’s evaluate the sign of each expression inside the absolute values for $ x \in [p, 15] $:

- $ x - p \geq 0 $, since $ x \geq p $, so $ |x - p| = x - p $
- $ x - 15 \leq 0 $, since $ x \leq 15 $, so $ |x - 15| = 15 - x $
- $ x - p - 15 \leq 0 $, since $ x \leq 15 $ and $ p > 0 $, so $ |x - p - 15| = p + 15 - x $

Substituting these into the function:

$$
f(x) = (x - p) + (15 - x) + (p + 15 - x)
$$

Simplify:

$$
f(x) = x - p + 15 - x + p + 15 - x = -x + 30
$$

So, on the interval $ [p, 15] $, the function simplifies to:

$$
f(x) = -x + 30
$$

---

### Step 3: Determine the Minimum of the Simplified Function

The function $ f(x) = -x + 30 $ is a **linear function** with a **negative slope**. Therefore, it is **strictly decreasing** on the interval $ [p, 15] $.

This means the **minimum value** occurs at the **right endpoint** of the interval, i.e., at $ x = 15 $.

---

### Step 4: Compute the Minimum Value

$$
f(15) = -15 + 30 = 15
$$

---

### Step 5: Confirm with an Alternative Approach

We can also interpret the function as the **sum of distances** from $ x $ to three points: $ p $, $ 15 $, and $ p + 15 $. The sum of absolute deviations is minimized at the **median** of the three points.

Since $ p < 15 < p + 15 $, the median is $ 15 $, and the minimum value of the function occurs at $ x = 15 $.

Evaluating the function at $ x = 15 $:

$$
f(15) = |15 - p| + |15 - 15| + |15 - p - 15| = (15 - p) + 0 + p = 15
$$

---

### Final Answer

$$
\boxed{15}
$$

Solution: Since $0 < p \le x \le 15,$ the absolute values simplify to \[f(x) = (x-p) - (x-15) - (x-p-15) = -x+30.\]The value of this expression is minimized when $x=15,$ giving $-15+30=\boxed{15}.$


Explanation:
Okay, so I need to find the equation of the line that consists of all vectors v such that the projection of v onto the vector (2, 1) is the zero vector. Hmm, let me recall how projections work.

First, the projection of a vector v onto another vector a is given by the formula:

proj_a v = ( (v · a) / ||a||² ) * a

Right? So if the projection is the zero vector, that means that (v · a) / ||a||² * a = 0. Since a is not the zero vector, the scalar multiple must be zero. Therefore, (v · a) must be zero.

So, in this case, the projection of v onto (2, 1) is zero if and only if v is orthogonal to (2, 1). Therefore, the set of all such vectors v is the set of vectors orthogonal to (2, 1).

But wait, the problem says that these vectors lie on a line. But the set of vectors orthogonal to a non-zero vector in R² is a line through the origin, right? Because in two dimensions, the orthogonal complement of a one-dimensional subspace is another one-dimensional subspace, which is a line.

So, if I can find the equation of the line that consists of all vectors orthogonal to (2, 1), that should be the answer.

Let me write that down. Let v = (x, y). Then, the dot product of v and (2, 1) is 2x + y. For them to be orthogonal, this must equal zero. Therefore:

2x + y = 0

So, solving for y, we get y = -2x.

Wait, but the problem says "the equation of this line in the form y = mx + b". So, in this case, m is -2 and b is 0. Therefore, the equation is y = -2x.

But let me check if I did everything correctly. Let me verify with an example. Suppose v is (1, -2). Then, the projection onto (2, 1) would be [( (1)(2) + (-2)(1) ) / (2² + 1²)] * (2, 1) = [ (2 - 2)/5 ] * (2, 1) = 0 * (2, 1) = (0, 0). So that works.

Another example: take v = (0, 0). The projection is obviously zero. That's on the line y = -2x since 0 = -2*0.

What about a vector not on the line? Let's say (1, 0). The projection would be [ (2*1 + 1*0)/5 ]*(2, 1) = (2/5)(2, 1) which is not zero. So that's correct.

So, the line is y = -2x. Therefore, the equation is y = -2x + 0, so m = -2 and b = 0.

But wait, the problem says "the set of vectors v" lie on a line. But vectors in R² can be represented as points in the plane, right? So, the line is y = -2x.

But let me think again. The set of vectors orthogonal to (2, 1) is a line through the origin, which is indeed y = -2x. So, that's the answer.

Alternatively, maybe I need to write it in terms of parametric equations? But the problem says "in the form y = mx + b", so y = -2x is correct.

Wait, but maybe I need to check if there's a different interpretation. The problem says "the set of vectors v such that proj_{(2,1)} v = 0". So, that's equivalent to v being in the orthogonal complement of (2,1). Which is a line through the origin. So, yes, the equation is y = -2x.

Alternatively, if I think of the projection formula again. Let me rederive it.

Let me write the projection formula again. For any vector v = (x, y), proj_{a} v = [(v · a)/||a||²] a.

So, if proj_{a} v = 0, then [(v · a)/||a||²] a = 0. Since a is non-zero, this implies that (v · a)/||a||² = 0, so v · a = 0.

Therefore, 2x + y = 0. So, solving for y gives y = -2x.

Therefore, the equation of the line is y = -2x.

So, the answer is y = -2x, which is in the form y = mx + b where m = -2 and b = 0.

Therefore, the equation is y = -2x.

I think that's correct. Let me check once more.

Suppose we have a vector (x, y) on the line y = -2x. Then, substituting into the dot product with (2, 1): 2x + y = 2x + (-2x) = 0. So, yes, that's correct.

Therefore, the line is y = -2x.

So, the answer is y = -2x.

**Final Answer**
The equation of the line is \boxed{y = -2x}.

To determine the equation of the line that contains all vectors $\mathbf{v}$ such that the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ is the zero vector, we begin by analyzing the condition:

$$
\operatorname{proj}_{\begin{pmatrix} 2 \\ 1 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}
$$

---

### Step 1: Understanding the Projection

The projection of a vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ onto another vector $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ is given by:

$$
\operatorname{proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} \mathbf{a}
$$

For this projection to be the zero vector, the scalar factor must be zero:

$$
\frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} = 0
$$

Since $\|\mathbf{a}\|^2 = 2^2 + 1^2 = 5 \neq 0$, the condition simplifies to:

$$
\mathbf{v} \cdot \mathbf{a} = 0
$$

---

### Step 2: Setting Up the Dot Product

Let $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$. Then:

$$
\mathbf{v} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix} = 2x + y
$$

Setting this equal to zero gives:

$$
2x + y = 0
$$

---

### Step 3: Solving for the Line Equation

Solving for $y$ in terms of $x$:

$$
y = -2x
$$

This is the equation of a line in the form $y = mx + b$, where $m = -2$ and $b = 0$.

---

### Final Answer

$$
\boxed{y = -2x}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of the line that consists of all vectors v such that the projection of v onto the vector (2, 1) is the zero vector. Hmm, let me recall how projections work.

First, the projection of a vector v onto another vector a is given by the formula:

proj_a v = ( (v · a) / ||a||² ) * a

Right? So if the projection is the zero vector, that means that (v · a) / ||a||² * a = 0. Since a is not the zero vector, the scalar multiple must be zero. Therefore, (v · a) must be zero.

So, in this case, the projection of v onto (2, 1) is zero if and only if v is orthogonal to (2, 1). Therefore, the set of all such vectors v is the set of vectors orthogonal to (2, 1).

But wait, the problem says that these vectors lie on a line. But the set of vectors orthogonal to a non-zero vector in R² is a line through the origin, right? Because in two dimensions, the orthogonal complement of a one-dimensional subspace is another one-dimensional subspace, which is a line.

So, if I can find the equation of the line that consists of all vectors orthogonal to (2, 1), that should be the answer.

Let me write that down. Let v = (x, y). Then, the dot product of v and (2, 1) is 2x + y. For them to be orthogonal, this must equal zero. Therefore:

2x + y = 0

So, solving for y, we get y = -2x.

Wait, but the problem says "the equation of this line in the form y = mx + b". So, in this case, m is -2 and b is 0. Therefore, the equation is y = -2x.

But let me check if I did everything correctly. Let me verify with an example. Suppose v is (1, -2). Then, the projection onto (2, 1) would be [( (1)(2) + (-2)(1) ) / (2² + 1²)] * (2, 1) = [ (2 - 2)/5 ] * (2, 1) = 0 * (2, 1) = (0, 0). So that works.

Another example: take v = (0, 0). The projection is obviously zero. That's on the line y = -2x since 0 = -2*0.

What about a vector not on the line? Let's say (1, 0). The projection would be [ (2*1 + 1*0)/5 ]*(2, 1) = (2/5)(2, 1) which is not zero. So that's correct.

So, the line is y = -2x. Therefore, the equation is y = -2x + 0, so m = -2 and b = 0.

But wait, the problem says "the set of vectors v" lie on a line. But vectors in R² can be represented as points in the plane, right? So, the line is y = -2x.

But let me think again. The set of vectors orthogonal to (2, 1) is a line through the origin, which is indeed y = -2x. So, that's the answer.

Alternatively, maybe I need to write it in terms of parametric equations? But the problem says "in the form y = mx + b", so y = -2x is correct.

Wait, but maybe I need to check if there's a different interpretation. The problem says "the set of vectors v such that proj_{(2,1)} v = 0". So, that's equivalent to v being in the orthogonal complement of (2,1). Which is a line through the origin. So, yes, the equation is y = -2x.

Alternatively, if I think of the projection formula again. Let me rederive it.

Let me write the projection formula again. For any vector v = (x, y), proj_{a} v = [(v · a)/||a||²] a.

So, if proj_{a} v = 0, then [(v · a)/||a||²] a = 0. Since a is non-zero, this implies that (v · a)/||a||² = 0, so v · a = 0.

Therefore, 2x + y = 0. So, solving for y gives y = -2x.

Therefore, the equation of the line is y = -2x.

So, the answer is y = -2x, which is in the form y = mx + b where m = -2 and b = 0.

Therefore, the equation is y = -2x.

I think that's correct. Let me check once more.

Suppose we have a vector (x, y) on the line y = -2x. Then, substituting into the dot product with (2, 1): 2x + y = 2x + (-2x) = 0. So, yes, that's correct.

Therefore, the line is y = -2x.

So, the answer is y = -2x.

**Final Answer**
The equation of the line is \boxed{y = -2x}.

To determine the equation of the line that contains all vectors $\mathbf{v}$ such that the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ is the zero vector, we begin by analyzing the condition:

$$
\operatorname{proj}_{\begin{pmatrix} 2 \\ 1 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}
$$

---

### Step 1: Understanding the Projection

The projection of a vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ onto another vector $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ is given by:

$$
\operatorname{proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} \mathbf{a}
$$

For this projection to be the zero vector, the scalar factor must be zero:

$$
\frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} = 0
$$

Since $\|\mathbf{a}\|^2 = 2^2 + 1^2 = 5 \neq 0$, the condition simplifies to:

$$
\mathbf{v} \cdot \mathbf{a} = 0
$$

---

### Step 2: Setting Up the Dot Product

Let $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$. Then:

$$
\mathbf{v} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix} = 2x + y
$$

Setting this equal to zero gives:

$$
2x + y = 0
$$

---

### Step 3: Solving for the Line Equation

Solving for $y$ in terms of $x$:

$$
y = -2x
$$

This is the equation of a line in the form $y = mx + b$, where $m = -2$ and $b = 0$.

---

### Final Answer

$$
\boxed{y = -2x}
$$

Solution:
Let $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}.$

From the formula of a projection,
\begin{align*}
\operatorname{proj}_{\begin{pmatrix} 2 \\ 1 \end{pmatrix}} \mathbf{v} &= \frac{\mathbf{v} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix}}{\left\| \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right\|^2} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\
&= \frac{\begin{pmatrix} x \\ y \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix}}{5} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\
&= \frac{2x + y}{5} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\
&= \begin{pmatrix} 0 \\ 0 \end{pmatrix}.
\end{align*}Then
\[\frac{2x + y}{5} = 0,\]so $2x + y = 0.$  Thus, the equation of the line is $\boxed{y = -2x}.$


Explanation:
Okay, so I need to find the modulus of this complex number w, which is given by (3z + 1) divided by (5z + 7), where z is 1 + i. Alright, let me start by recalling that the modulus of a complex number a + bi is sqrt(a² + b²). But first, I need to figure out what w actually is. Since z is given, maybe I should substitute z into the expression for w and simplify it first. Then, once I have w in the form a + bi, I can compute its modulus.

Let me write down the given values:

z = 1 + i

w = (3z + 1)/(5z + 7)

So, first step: compute the numerator and the denominator separately.

Starting with the numerator: 3z + 1.

Since z = 1 + i, multiplying by 3 gives 3*(1 + i) = 3 + 3i. Then adding 1: 3 + 3i + 1 = 4 + 3i. So the numerator is 4 + 3i.

Now the denominator: 5z + 7. Let me compute that. 5*(1 + i) = 5 + 5i. Then adding 7: 5 + 5i + 7 = 12 + 5i. So the denominator is 12 + 5i.

Therefore, w = (4 + 3i)/(12 + 5i). Now, to simplify this complex fraction, I need to rationalize the denominator. That is, multiply numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 12 + 5i is 12 - 5i. So let me do that.

Multiply numerator and denominator by 12 - 5i:

w = [(4 + 3i)(12 - 5i)] / [(12 + 5i)(12 - 5i)]

First, compute the denominator. Let me recall that (a + b)(a - b) = a² - b². So here, a is 12, b is 5i. Therefore, denominator is 12² - (5i)². Let me compute that:

12² is 144. (5i)² is 25i², which is 25*(-1) = -25. Therefore, denominator is 144 - (-25) = 144 + 25 = 169. Okay, so denominator is 169. That's straightforward.

Now the numerator: (4 + 3i)(12 - 5i). Let me expand this using distributive property (FOIL method):

First: 4*12 = 48

Outer: 4*(-5i) = -20i

Inner: 3i*12 = 36i

Last: 3i*(-5i) = -15i²

Now combine all these terms:

48 - 20i + 36i -15i²

Combine like terms:

The real parts: 48 and -15i². Wait, but i² is -1, so -15i² is -15*(-1) = +15. So the real parts are 48 + 15 = 63.

The imaginary parts: -20i + 36i = 16i.

Therefore, the numerator simplifies to 63 + 16i.

So putting it all together, w = (63 + 16i)/169. Which can be written as (63/169) + (16/169)i.

Now, to find |w|, which is the modulus of w. Since w is a complex number a + bi, modulus is sqrt(a² + b²). So here, a is 63/169 and b is 16/169.

Therefore, |w| = sqrt[(63/169)² + (16/169)²] = sqrt[(63² + 16²)/169²] = sqrt(63² + 16²)/169.

Let me compute 63² and 16²:

63²: Let me compute 60² + 2*60*3 + 3² = 3600 + 360 + 9 = 3969. Wait, but 63*63: 60*60=3600, 60*3=180, 3*60=180, 3*3=9. So (60+3)^2 = 60² + 2*60*3 + 3² = 3600 + 360 + 9 = 3969. Correct.

16² is 256.

So adding them: 3969 + 256. Let me compute that. 3969 + 200 = 4169, then +56 = 4225. Wait, 3969 + 256:

Let me check again:

3969 + 256:

Breakdown 256 into 200 + 50 + 6.

3969 + 200 = 4169

4169 + 50 = 4219

4219 + 6 = 4225. Yes, that's correct. 4225.

So sqrt(4225)/169. What is sqrt(4225)? Let me think. 65² is 4225 because 60²=3600, 70²=4900, so 65² = (60 + 5)^2 = 60² + 2*60*5 + 5² = 3600 + 600 + 25 = 4225. Yes! So sqrt(4225) is 65.

Therefore, |w| = 65 / 169. Simplify that fraction. Let me see if 65 and 169 have common factors. 65 factors into 5*13. 169 is 13². So 65 is 5*13, so divide numerator and denominator by 13: 65 ÷13=5, 169 ÷13=13. Therefore, 5/13.

So |w| is 5/13. Let me check my steps again to make sure I didn't make a mistake.

First, z = 1 + i. Then 3z + 1: 3*(1 + i) + 1 = 3 + 3i + 1 = 4 + 3i. Correct. Then 5z + 7: 5*(1 + i) + 7 = 5 + 5i + 7 = 12 + 5i. Correct. Then multiplying numerator and denominator by 12 - 5i. Denominator becomes 12² - (5i)^2 = 144 - (-25) = 169. Correct. Numerator: (4 + 3i)(12 - 5i). Let me check that again:

First term: 4*12 = 48. Second term: 4*(-5i) = -20i. Third term: 3i*12 = 36i. Fourth term: 3i*(-5i) = -15i² = +15. Then combining real parts: 48 + 15 = 63. Imaginary parts: -20i + 36i = 16i. So numerator is 63 + 16i. Correct. Then modulus squared is (63² + 16²)/169². Which is 4225 / (169²). Then sqrt(4225)/169 = 65/169 = 5/13. Correct. So yes, seems like the answer is 5/13.

Alternatively, maybe there's a smarter way to compute |w| without computing w explicitly? Let me think. Since |w| = |3z + 1| / |5z + 7|. Because modulus of a quotient is the quotient of the moduli. So maybe I can compute |3z + 1| and |5z + 7| separately and then divide them. Let me try this method to verify.

First, compute |3z + 1|. Since z = 1 + i, 3z + 1 is 4 + 3i as before. The modulus is sqrt(4² + 3²) = sqrt(16 + 9) = sqrt(25) = 5. Then compute |5z + 7|. 5z + 7 is 12 + 5i, modulus is sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13. Therefore, |w| = 5 / 13. Same result! So that's a quicker way. So that confirms the answer is 5/13. So either way, whether I compute w explicitly and then take modulus, or use the property of modulus of quotient, I get the same answer. Therefore, I can be confident that the answer is 5/13.

But just to make sure, let me check once again. Let me compute |3z + 1|:

z = 1 + i. So 3z = 3 + 3i. Then 3z + 1 = 4 + 3i. The modulus is sqrt(4² + 3²) = 5. Correct. Then |5z + 7|: 5z = 5 + 5i. 5z + 7 = 12 + 5i. Modulus sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13. Correct. Therefore, |w| = 5/13. Yep, that's straightforward. So that's the answer.

Alternatively, if I didn't remember that property, but I could have used the formula |a/b| = |a| / |b|. So that's a useful property. So maybe that's the better way here, since it avoids dealing with complex fractions. But either way, same result.

So, conclusion: |w| is 5/13.

**Final Answer**
The modulus of $ w $ is \boxed{\dfrac{5}{13}}.

To find the modulus of the complex number $ w = \dfrac{3z + 1}{5z + 7} $, where $ z = 1 + i $, we proceed step by step using properties of complex numbers and their moduli.

---

### Step 1: Compute the Numerator and Denominator

Given:
$$
z = 1 + i
$$

Compute the numerator:
$$
3z + 1 = 3(1 + i) + 1 = 3 + 3i + 1 = 4 + 3i
$$

Compute the denominator:
$$
5z + 7 = 5(1 + i) + 7 = 5 + 5i + 7 = 12 + 5i
$$

So,
$$
w = \frac{4 + 3i}{12 + 5i}
$$

---

### Step 2: Use the Modulus Property of Quotients

Recall the important property of complex numbers:
$$
\left| \frac{a}{b} \right| = \frac{|a|}{|b|}
$$

Apply this to $ w $:
$$
|w| = \frac{|3z + 1|}{|5z + 7|}
$$

We already computed:
- $ |3z + 1| = |4 + 3i| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $
- $ |5z + 7| = |12 + 5i| = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 $

Thus:
$$
|w| = \frac{5}{13}
$$

---

### Final Answer

$$
\boxed{\dfrac{5}{13}}
$$

[Assistant]
Explanation:
Okay, so I need to find the modulus of this complex number w, which is given by (3z + 1) divided by (5z + 7), where z is 1 + i. Alright, let me start by recalling that the modulus of a complex number a + bi is sqrt(a² + b²). But first, I need to figure out what w actually is. Since z is given, maybe I should substitute z into the expression for w and simplify it first. Then, once I have w in the form a + bi, I can compute its modulus.

Let me write down the given values:

z = 1 + i

w = (3z + 1)/(5z + 7)

So, first step: compute the numerator and the denominator separately.

Starting with the numerator: 3z + 1.

Since z = 1 + i, multiplying by 3 gives 3*(1 + i) = 3 + 3i. Then adding 1: 3 + 3i + 1 = 4 + 3i. So the numerator is 4 + 3i.

Now the denominator: 5z + 7. Let me compute that. 5*(1 + i) = 5 + 5i. Then adding 7: 5 + 5i + 7 = 12 + 5i. So the denominator is 12 + 5i.

Therefore, w = (4 + 3i)/(12 + 5i). Now, to simplify this complex fraction, I need to rationalize the denominator. That is, multiply numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 12 + 5i is 12 - 5i. So let me do that.

Multiply numerator and denominator by 12 - 5i:

w = [(4 + 3i)(12 - 5i)] / [(12 + 5i)(12 - 5i)]

First, compute the denominator. Let me recall that (a + b)(a - b) = a² - b². So here, a is 12, b is 5i. Therefore, denominator is 12² - (5i)². Let me compute that:

12² is 144. (5i)² is 25i², which is 25*(-1) = -25. Therefore, denominator is 144 - (-25) = 144 + 25 = 169. Okay, so denominator is 169. That's straightforward.

Now the numerator: (4 + 3i)(12 - 5i). Let me expand this using distributive property (FOIL method):

First: 4*12 = 48

Outer: 4*(-5i) = -20i

Inner: 3i*12 = 36i

Last: 3i*(-5i) = -15i²

Now combine all these terms:

48 - 20i + 36i -15i²

Combine like terms:

The real parts: 48 and -15i². Wait, but i² is -1, so -15i² is -15*(-1) = +15. So the real parts are 48 + 15 = 63.

The imaginary parts: -20i + 36i = 16i.

Therefore, the numerator simplifies to 63 + 16i.

So putting it all together, w = (63 + 16i)/169. Which can be written as (63/169) + (16/169)i.

Now, to find |w|, which is the modulus of w. Since w is a complex number a + bi, modulus is sqrt(a² + b²). So here, a is 63/169 and b is 16/169.

Therefore, |w| = sqrt[(63/169)² + (16/169)²] = sqrt[(63² + 16²)/169²] = sqrt(63² + 16²)/169.

Let me compute 63² and 16²:

63²: Let me compute 60² + 2*60*3 + 3² = 3600 + 360 + 9 = 3969. Wait, but 63*63: 60*60=3600, 60*3=180, 3*60=180, 3*3=9. So (60+3)^2 = 60² + 2*60*3 + 3² = 3600 + 360 + 9 = 3969. Correct.

16² is 256.

So adding them: 3969 + 256. Let me compute that. 3969 + 200 = 4169, then +56 = 4225. Wait, 3969 + 256:

Let me check again:

3969 + 256:

Breakdown 256 into 200 + 50 + 6.

3969 + 200 = 4169

4169 + 50 = 4219

4219 + 6 = 4225. Yes, that's correct. 4225.

So sqrt(4225)/169. What is sqrt(4225)? Let me think. 65² is 4225 because 60²=3600, 70²=4900, so 65² = (60 + 5)^2 = 60² + 2*60*5 + 5² = 3600 + 600 + 25 = 4225. Yes! So sqrt(4225) is 65.

Therefore, |w| = 65 / 169. Simplify that fraction. Let me see if 65 and 169 have common factors. 65 factors into 5*13. 169 is 13². So 65 is 5*13, so divide numerator and denominator by 13: 65 ÷13=5, 169 ÷13=13. Therefore, 5/13.

So |w| is 5/13. Let me check my steps again to make sure I didn't make a mistake.

First, z = 1 + i. Then 3z + 1: 3*(1 + i) + 1 = 3 + 3i + 1 = 4 + 3i. Correct. Then 5z + 7: 5*(1 + i) + 7 = 5 + 5i + 7 = 12 + 5i. Correct. Then multiplying numerator and denominator by 12 - 5i. Denominator becomes 12² - (5i)^2 = 144 - (-25) = 169. Correct. Numerator: (4 + 3i)(12 - 5i). Let me check that again:

First term: 4*12 = 48. Second term: 4*(-5i) = -20i. Third term: 3i*12 = 36i. Fourth term: 3i*(-5i) = -15i² = +15. Then combining real parts: 48 + 15 = 63. Imaginary parts: -20i + 36i = 16i. So numerator is 63 + 16i. Correct. Then modulus squared is (63² + 16²)/169². Which is 4225 / (169²). Then sqrt(4225)/169 = 65/169 = 5/13. Correct. So yes, seems like the answer is 5/13.

Alternatively, maybe there's a smarter way to compute |w| without computing w explicitly? Let me think. Since |w| = |3z + 1| / |5z + 7|. Because modulus of a quotient is the quotient of the moduli. So maybe I can compute |3z + 1| and |5z + 7| separately and then divide them. Let me try this method to verify.

First, compute |3z + 1|. Since z = 1 + i, 3z + 1 is 4 + 3i as before. The modulus is sqrt(4² + 3²) = sqrt(16 + 9) = sqrt(25) = 5. Then compute |5z + 7|. 5z + 7 is 12 + 5i, modulus is sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13. Therefore, |w| = 5 / 13. Same result! So that's a quicker way. So that confirms the answer is 5/13. So either way, whether I compute w explicitly and then take modulus, or use the property of modulus of quotient, I get the same answer. Therefore, I can be confident that the answer is 5/13.

But just to make sure, let me check once again. Let me compute |3z + 1|:

z = 1 + i. So 3z = 3 + 3i. Then 3z + 1 = 4 + 3i. The modulus is sqrt(4² + 3²) = 5. Correct. Then |5z + 7|: 5z = 5 + 5i. 5z + 7 = 12 + 5i. Modulus sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13. Correct. Therefore, |w| = 5/13. Yep, that's straightforward. So that's the answer.

Alternatively, if I didn't remember that property, but I could have used the formula |a/b| = |a| / |b|. So that's a useful property. So maybe that's the better way here, since it avoids dealing with complex fractions. But either way, same result.

So, conclusion: |w| is 5/13.

**Final Answer**
The modulus of $ w $ is \boxed{\dfrac{5}{13}}.

To find the modulus of the complex number $ w = \dfrac{3z + 1}{5z + 7} $, where $ z = 1 + i $, we proceed step by step using properties of complex numbers and their moduli.

---

### Step 1: Compute the Numerator and Denominator

Given:
$$
z = 1 + i
$$

Compute the numerator:
$$
3z + 1 = 3(1 + i) + 1 = 3 + 3i + 1 = 4 + 3i
$$

Compute the denominator:
$$
5z + 7 = 5(1 + i) + 7 = 5 + 5i + 7 = 12 + 5i
$$

So,
$$
w = \frac{4 + 3i}{12 + 5i}
$$

---

### Step 2: Use the Modulus Property of Quotients

Recall the important property of complex numbers:
$$
\left| \frac{a}{b} \right| = \frac{|a|}{|b|}
$$

Apply this to $ w $:
$$
|w| = \frac{|3z + 1|}{|5z + 7|}
$$

We already computed:
- $ |3z + 1| = |4 + 3i| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $
- $ |5z + 7| = |12 + 5i| = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 $

Thus:
$$
|w| = \frac{5}{13}
$$

---

### Final Answer

$$
\boxed{\dfrac{5}{13}}
$$

Solution: Plugging in, we have $w = \dfrac{3(1+i)+1}{5(1+i)+7} = \dfrac{4+3i}{12+5i}$. We could write this in the form $a+bi$ and take the magnitude, but it's easier to use the fact that, for all complex numbers $a$ and $b$, $\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}$. The magnitude of the numerator is $\sqrt{3^2+4^2} = \sqrt{25} = 5$, and the magnitude of the denominator is $\sqrt{12^2 + 5^2} = \sqrt{169} = 13$. So $|w| = \boxed{\frac{5}{13}}$.