Related Rates: Baseball Diamond

Question

A baseball diamond is a square with sides length 90 ft. A batter runs towards the first base with a speed of 20 ft/sec.

a) At what rate is his distance from second base changing when he is halfway to first base.

b) At what rate is his distance from third base changing at the same moment?

---

I'm not sure how to set up this problem at all. I know whatever expression I have, I must evaluate its derivative at the moment $t=9/4$ seconds because that is the time it takes to reach halfway to the first base.

Answer 1

Set up a coordinate system. You can place home at the origin and first base at $(90,0)$. Where are second and third? Where is halfway to first? What is the distance from there to second? To third? Write an equation for the distance of the runner to second as a function of $t$ and take the derivative.

Answer 2

If your speed is $20 \frac{feet}{second}$, your rate of change is $20fps$. We will call that $r=20$

Your distance from first base is $90ft$. So this should be the equation for distance left to first base:

$$D_1(t)=90-rt$$

Plugging in $r=20$ and evaluating the equation at $t=9/4$:

$$D_1\left(\frac{9}{4}\right)=90-20\left(\frac{9}{4}\right) = 45$$

So it looks like this equation checks out. Now, using the concepts from above, you need two make two equations $D_2(t)$ and $D_3(t)$ to evaluate distance from second and distance from third. Your thinking is correct that you will need to find the derivative of both equations.

Answer 3

$D$ is the distance from 3rd base.

$D = \sqrt {x^2+90^2}$

$\frac {dD}{dt} = \frac {x}{\sqrt {x^2+90^2}} \frac {dx}{dt}$

$\frac {dx}{dt} = 20 \frac {ft}{s}$

When the runner is half-way, $x = 45$

That is enough information to solve for $\frac {dD}{dt}$

We can argue by symmetry that when the runner is half way, he is moving toward second base at the same speed he is moving away from first base.

Or you if you don't like that arguement, you can work though the calculus only using the distance from second base as $\sqrt {(90-x)^2 + 90^2}$