AP Calculus - Minimizing Time Traveled

Date: 12/22/98 at 11:54:51
From: Heidi Mergenthaler
Subject: AP Calculus

Dear Dr. Math,

Here is a problem that our AP Calculus teacher did not know how to 
solve:

A person on a boat in a lake is 9 km from the shore and must go to a 
point 12 km down the shoreline in the shortest possible time. The 
person can walk 8 km per hour and the boat can travel r km per hour.

a) Assume that the person should travel by boat and by foot. Let d be 
the distance down the shoreline the person should strike land for the 
shortest total travel time. Write d as a function of r.

b) Sketch the graph of the function d(r). Determine the slowest speed 
of the boat so the shortest possible time criterion is met by making 
the entire trip by water. Use this to give the domain of the function 
in the context of this problem.

c) Determine the concavity of the graph of the function d over the 
domain stated in part b. What information does the concavity give about 
the relation between d and r?

Our class attempted this problem and wrote an equation based on the 
given information. We took the derivative of our equation to find the 
answer to part a, but the answer we arrived at was different from the 
answer given in the book. Our answer was d=(r-36)/4, from the equation 
d(r) = r * sqrt(d^2+81) + 8(12-d). As stated, d is the distance down 
the shoreline where the person should strike land. Thus 12-d is the 
distance from the point where the person lands to his or her 
destination. Using the Pythagorean Theorem, we found the distance from 
the boat to the point where the person will land to be the square root 
of (d^2 + 81). The answer the book gave us is d(r) = 9r/sqrt(64-r^2). 

We can't figure out what we did wrong, and we hope you will be able to 
help us.

Date: 12/23/98 at 13:45:09
From: Doctor Nick
Subject: Re: AP Calculus

Hi Heidi -

I think I know where your problem lies. Your equation for d(r) is where 
your problem starts.

Remember that d = r * t, and the first thing to notice in the question 
is that we're writing d in terms of rate, not rate and time, so we need 
to get rid of the time component. First find the total time of the 
trip. The time for the trip is t = sqrt(d^2+81)/r + (12-d)/8. What you 
want to do is minimize this function to find the shortest total travel 
time. You need to differentiate this with respect to d, set it equal to 
zero, and solve for d. This will give you a function which depends on 
r, which is d(r). If you do this, you'll get d(r) = 9r/sqrt(64-r^2), as 
your book says. Remember that this function results in the shortest 
overall travel time.

I'll give you some hints for the rest of the problem. For (b), solve 
for r when d = 12. Why would you want to do this? Try plugging in a 
larger r than the r you get. You'll see that the resulting d is farther 
down the shore than you want. So this r defines the domain of the 
function d(r).

For (c), find the second derivative. You'll see that as r increases, 
d increases. Why does this make sense? Well, as the rate of the boat 
increases (i.e.: the boat goes faster), you'll want to stay in the boat 
longer and hit land farther down the shore. If the boat is slow, you'll 
want to switch to walking sooner. 

I hope this helps get you on the right track. Write back if you need 
more help.

- Doctor Nick, The Math Forum
  http://mathforum.org/dr.math/