# [ASK] Minimum Length of AP + PB

• MHB
• Monoxdifly
In summary: The problem is not clear and could be interpreted in multiple ways. However, the most logical interpretation would be that APB is a triangle and the minimum length of AP + PB would be when P is the same point as B, which has coordinates (10,0). Therefore, in summary, the minimum length of AP + PB occurs when point P has coordinates (10,0).
Monoxdifly
MHB
The point A is located on the coordinate (0, 5) and B is located on (10, 0). Point P(x, 0) is located on the line segment OB with O(0, 0). The coordinate of P so that the length AP + PB minimum is ...
A. (3, 0)
B. (3 1/4, 0)
C. (3 3/4, 0)
D. (4 1/2, 0)
E. (5, 0)

What I did:
f(x) = AP + PB =$$\displaystyle \sqrt{5^2+x^2}+(10-x)=\sqrt{25+x^2}+10-x$$
In order to make AP + PB minimum, so:
f'(x) = 0
$$\displaystyle \frac12(25+x^2)^{-\frac12}(2x)+(-1)=0$$
$$\displaystyle \frac{x}{\sqrt{25+x^2}}=1$$
$$\displaystyle x=\sqrt{25+x^2}$$
$$\displaystyle x^2=25+x^2$$
This is where I got stuck. Subtracting $$\displaystyle x^2$$ from both sides would leave me with 0 = 25 which is obviously incorrect. Where did I do wrong?

uhh ...

$\dfrac{x}{\sqrt{x^2+25}} < 1$ for any value of $x$

So, which steps should I fix? And become what?

There is something wrong with this question. The length AP + PB will be a minimum when APB is a straight line, and that happens when P = B. So the answer should be that P has coordinates (10,0).

Notice that to allow for the possibility $x>10$, the formula for $f(x)$ should be $\sqrt{5^2+x^2} + |10-x|$, which is indeed minimised at $x=10$.

Opalg said:
There is something wrong with this question. The length AP + PB will be a minimum when APB is a straight line, and that happens when P = B. So the answer should be that P has coordinates (10,0).
Come to think of it, you're right. APB is a triangle thus the minimum length of AP + PB should be AB. Thanks. Glad I'm not the one who messed up.

APB is a triangle ...

The problem statement does not say APB is a triangle. I believe the problem's aim was to show the possibility of an endpoint minimum.

I think I should have said that APB is "supposed to be" a triangle, not necessarily a triangle itself.

## What is the minimum length of AP + PB?

The minimum length of AP + PB is determined by the distance between points A and B, as well as the angle formed by the two points. This can be calculated using the Pythagorean theorem and trigonometric functions.

## How is the minimum length of AP + PB calculated?

The minimum length of AP + PB is calculated by finding the distance between points A and B, and then using trigonometric functions to determine the angle formed by the two points. This information is then used in the Pythagorean theorem to find the minimum length.

## Why is the minimum length of AP + PB important?

The minimum length of AP + PB is important because it can be used to determine the shortest possible distance between two points, which is useful in many scientific and engineering applications. It can also help in optimizing routes and minimizing travel time.

## What factors influence the minimum length of AP + PB?

The minimum length of AP + PB is influenced by the distance between points A and B, as well as the angle formed by the two points. Other factors such as terrain, obstacles, and speed can also affect the minimum length.

## Can the minimum length of AP + PB be negative?

No, the minimum length of AP + PB cannot be negative. It represents a physical distance between two points and therefore must be a positive value. If the calculated value is negative, it indicates an error in the calculations.

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