1. ## word problems

I actually have three but i put them up one at a time.
The first:
A truck is 250 mi due east of a sports car and is traveling west at a constant speed of 60mi/h. Meanwhile, the sports car is going north at 80mi/h. When will the truck and the car be closest to each other? what is the minimum distance between them? Hint: Minimize the square of the distance. The answer says The minimum distance is 200 mi.

I figure this is a Pythag. theorem problem. the adjacent(the truck) is heading left at 60mi/h. The opposite(the car) is going up at 80mi/h. and hypot. is 250mi. I tried deriving to get x or y, but that did not work. Please help.

2. By Pythagoras:

$D^2=(250-60t)^{2}+(80t)^{2}$

$D^2=t^{2}-3t+\frac{25}{4}$

Now, minimize.

3. Now, I am lost, do I try to find the derivitive next?

4. Yes, find the derivative, set to 0 and solve for t. Plug it back into the original equation. You will get the answer you
seek.

You can just differentiate $t^{2}-3t+\frac{25}{4}$

5. I did what u said and got the right answer. Thanks. I am just a little bit confused on how (250-60t)^2+(80t)^2 is (or at least simplified to)t^2-3t+25/4? Other than that, i'm good on this problem.

6. All I did was expand and divide through by 10000.

See, when you expand you get $10000t^{2}-30000t+62500$

I divided by 10000 to simplify.

7. I expanded and, and i got 3600t^2-30000t+68900. I exactly pressed the equation right and it gave that. I am not sure what is happening here.

8. Originally Posted by driver327
I did what u said and got the right answer. Thanks. I am just a little bit confused on how (250-60t)^2+(80t)^2 is (or at least simplified to)t^2-3t+25/4? Other than that, i'm good on this problem.
Originally Posted by driver327
I expanded and, and i got 3600t^2-30000t+68900. I exactly pressed the equation right and it gave that. I am not sure what is happening here.
$(250-60t)^2+(80t)^2$

$= 3600t^2 - 30000t +62500 + 6400t^2$

$= 10000t^2 - 30000t +62500$

-Dan