Baseball diamond word problem

This isn't a homework problem. I'm trying to go through Calculus and Analytic Geometry by Rodin on my own and I'm already stumped. Here's the problem:

A baseball diamond is a square whose sides are 90 feet. Suppose a man runs around the bases at a speed of 20 feet per second, starting from home plate. Express the shortest straight line distance f between this man and home plate as a function of the time t in seconds since he started running. [Hint: Four equations will be needed.]

This should be easy but I just can't do it. :(

If it matters, I am fine with a hint or a number of hints. I'd prefer to find the answer with some help than to be given the answer outright. So far I've written out a table that gives the horizontal distance, the vertical distance, and the sum of the horizontal and vertical distances for given values of t. I'm not sure how to "mod out" the 90 feet, if that makes sense.

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Originally Posted by Poophead

This isn't a homework problem. I'm trying to go through Calculus and Analytic Geometry by Rodin

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Do you mean "Rudin"? Good book but not an easy one on your own.

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on my own and I'm already stumped. Here's the problem:

A baseball diamond is a square whose sides are 90 feet. Suppose a man runs around the bases at a speed of 20 feet per second, starting from home plate. Express the shortest straight line distance f between this man and home plate as a function of the time t in seconds since he started running. [Hint: Four equations will be needed.]

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For problems like this the first thing you should do is set up a coordinate system so that you can write equations. Set up your coordinate system so that the origin is at home plate and the positive x-axis extends to the first base. Each "unit" will be one foot.

(There are many possible coordinate systems. My first thought was the "obvious" origin at home plate, y-axis straight up through second base, but I think this will be simpler.)

While the runner goes from home plate to first base, at 20 feet per second, his distance from home plate is just his "y" coordinate. Once he turns toward second base, he is running along the line x= 90 so his "position" is (90, y). What is the distance from that to home plate?

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This should be easy but I just can't do it. :(

If it matters, I am fine with a hint or a number of hints. I'd prefer to find the answer with some help than to be given the answer outright. So far I've written out a table that gives the horizontal distance, the vertical distance, and the sum of the horizontal and vertical distances for given values of t. I'm not sure how to "mod out" the 90 feet, if that makes sense.

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Okay, I think I misunderstood the problem. When it said I would need four equations, I thought that meant that I would need to find four equations on my way to finding f. But now I think that I'm supposed to define f in a piecewise manner, which makes the whole problem rather easy. So here's what I've got:

0 <= t <= 4 1/2: f(t) = 20t

4 1/2 < t <= 9: f(t) = sqrt(8100 + [20 * (t - 4 1/2)]^2)

9 < t <= 13 1/2: f(t) = sqrt((90 - 20(t - 9))^2 + 8100)

13 1/2 < t <= 18: f(t) = 90 - 20(t - 13 1/2)

If that's right (or close), I was waaaaaay overthinking the problem.