# Thread: I solved this derivative word problem, but what does it really mean?

1. ## I solved this derivative word problem, but what does it really mean?

I have a word problem which I did all the arithmetic for but what do I do with the x I solved for? Here it is:

A man is on the bank river, which is 1 mile wide. He wants to travel to a town on the opposite bank, but 1 mile upstream. He intends to row on a straight line to point p on the opposite bank and then walk the remaining distance along the bank. To what point should he row in order to reach his destination in the least amount of time if he can walk 5 miles per hour and row 3 miles per hour?

Let x = distance between the point directly across on the opposite bank and the point where he needs to row.
s = distance he will row.
t = time spent rowing and walking.

t = s/3 + (1 - x)/5
Using Pythagorean theorem, we can express s in terms of x as
s^2 = 1 + x^2
s = sqrt(1+ x^2)

t = sqrt(1+ x^2) / 3 + (1 - x) / 5
Taking the derivative of t with respect to x.
dt/dx = (x / 3(sqrt(1 + x^2) - 1/5

Setting dt/dx = 0 and solving for x,
(x / 3(sqrt(1 + x^2) - 1/5 = 0
(x / 3(sqrt(1 + x^2) = 1/5
5x / sqrt(1 + x^2) = 3
5x = 3(sqrt(1 + x^2))
25x^2 = 9(1 + x^2)
25x^2 = 9 + 9x^2
16x^2 = 9
4x = 3
x = 3/4

2. What a strange place to get stuck. That was a lot of work to have it fall apart at the end.

Read the problem statement. "To what point should he row "
Read your definition: "x = distance between the point directly across on the opposite bank and the point where he needs to row."

It may have helped if you had included one more explanatory note to your defintiion section.

1-x = Distance he will walk. You have that in your formulation and solution, but you didn't say it explicitly.