Solved: Related Rates Problem with Baseball Diamond Distance and Speed

[bfr]

[SOLVED] Simple Related Rates

Homework Statement

A baseball diamond is a square 90 ft on a side. A runner travels from home plate to first base at 20 ft/sec. How fast is the runner's distance changing when the runner is half way to first base?

Homework Equations

a^2+b^2=c^2

The Attempt at a Solution

The distance from the runner to second base is: D=sqrt((20t)^2+90^2)

dD/dt=(40t)/(sqrt(4x^2+81))

The runner is half way to second base when... 20t=45; t=9/4;

I plug t into dD/dt and get 4*sqrt(5)~=8.94. But wait...shouldn't dD/dt be negative, because D is decreasing, because the runner is getting closer to second base?

EDIT: I got it. Distance between runner and first base=r=90-20t. Distance between runner and second base=D=sqrt(r^2+90^2)

dD/dt=(1/2)(r^2+90^2)^(-1/2) * (2r*dr/dt)

Like I said, the runner is half way to second base when 20t=45;t=9/4.

dr/dt=-20

r=90-20(9/4)

Plug r and dr/dt in, and get -4*sqrt(5), which I'm pretty sure is the correct answer.

 

[dynamicsolo]

The problem you seem to be solving appears to be different from what the question asks. Is the runner going from home plate to first base, or first base to second base? What distance is being measured, from the runner to home plate, to first base, or to second base?

Your work looks like you are finding distance from home plate in one section, then from second base in a different portion. Could you check to see what the problem is calling for and what it is you are solving?

 

[HallsofIvy]

Did you leave out part of the problem? You say

A baseball diamond is a square 90 ft on a side. A runner travels from home plate to first base at 20 ft/sec. How fast is the runner's distance changing when the runner is half way to first base?

Distance to what? In your work you seem to be assuming that the question is about the runner's distance to second base but you don't say that!

By the way, when working with distances, it is better not to take the square root: D²= (20t)²+ (90)². Now just use "implicit differentiation": 2D D'= 4(20t)(20)