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/
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