Optimization CALCULUS 1

Question

Brandon is on one side of a river that is 80 m wide and wants to reach a point 200 m downstream on the opposite side as quickly as possible by swimming diagonally across the river and then running the rest of the way. Find the minimum amount of time if Brandon can swim at 2 m/s and run at 5 m/s.

Answer 1

OK. No current. That would make a difference in an optimization problem. The idea behind optimization is that you want to find the best, the largest, the smallest of something among a number (maybe an infinite number) of possibilities. Yikes!

 

In this case, your goal is to minimize Brandon's total travel time. And we know the total distance he travels. What we don't know is how he splits it.

 

So if we go with time = distance / rate, then the time he spends swimming is

 

(80² + x²)^(1/2) / 2

 

and the time he spends running is

 

(200 - x) / 5

 

where in both of these cases x = the distance Brandon swims *along the direction of the river*.

 

f(x), the thing we want to minimize, is the total time.

 

f(x) = [(6400 + x²)^(1/2) / 2] + (200 - x)/5

 

To minimize f(x), find f'(x), set it to zero, and solve for x.

 

f'(x) = [(6400 + x²)^(-1/2)(2x) / 2] - (1/5) = 0

 

Heck with calculating that by hand. I used Wolfram Alpha. And I got

 

x = 20 (2/3)^(1/2)

which is somewhere between 16 and 17 m. That's how far in the direction of the river Brandon swam.

 

Then I used Wolfram Alpha again to plug x into the equation for f(x) and found Brandon's travel time was about 77.6 seconds.

You can get these solutions by hand, but I really think that isn't what's being taught here. You want to be able to set up the problem.

Sanity check: Brandon can't run the whole way. If he swims straight across the river, then runs the whole 200 m downstream, his travel time is 140 sec. He CAN swim the whole way, a distance of 215 m. His travel time then is 107.8 sec. The solution above is something between the two extremes and gives a much better time.