Thread: Finding the distance between two points

1. Finding the distance between two points

Let Q = (0,5) and R = (10,6) be given points in the plane. We want to find the point P = (x,0) on the x axis such that the sum of distances PQ+PR is as small as possible.
To solve this problem, we need to minimize the following function of x: f(x) = ?
over the closed interval [A,B] where A = ? and B = ?

2. Originally Posted by derekjonathon
Let Q = (0,5) and R = (10,6) be given points in the plane. We want to find the point P = (x,0) on the x axis such that the sum of distances PQ+PR is as small as possible.
To solve this problem, we need to minimize the following function of x: f(x) = ?
over the closed interval [A,B] where A = ? and B = ?
$PQ = d_1$

$d_1 = \sqrt{(x-0)^2 + (0-5)^2}$

$PR = d_2$

$d_2 = \sqrt{(x-10)^2 + (0-6)^2}$

$d_1 + d_2 = S$

find $\frac{dS}{dx}$ and minimize

3. Still not getting it...

Thank you for your help, I really appreciate it...

But I am still not getting what the technique is here to answer the question.

I understand finding distance (d1) and distance (d2)

but the derivative part has me thrown off. Also, I don't get what function is supposed to be minimized...

How do I go step by step through this?

4. Originally Posted by derekjonathon
Thank you for your help, I really appreciate it...

But I am still not getting what the technique is here to answer the question.

I understand finding distance (d1) and distance (d2)

but the derivative part has me thrown off. Also, I don't get what function is supposed to be minimized...

How do I go step by step through this?
minimize the function $S = \sqrt{x^2+25} + \sqrt{x^2-20x+136}$

start by finding $\frac{dS}{dx}$, set the result equal to 0, and solve for the value of x that minimizes S.