Optimize Rower's Trip: Find Min. Time From P to R

[joess]

Homework Statement

A rower in a boat at a point P, 3km from the closest point Q on a straight shorline, wishes to reach a point r which is 5 km along the shoreline from Q. If he can row at 2km/h and walk at 4km/h along the shoreline, how far from point Q should the rower land the boat in order that the total time for the trip PR is minimized.

The Attempt at a Solution

Okay so if I let x be the distance from point Q, then the equation should be:

t = 2(x^2+9)^(1/2) + 4(5-x)

right?

Then

dt/dx = 2x/(x^2+9)^(1/2) - 4
0 = 2x/(x^2+9)^(1/2) - 4
4(x^2+9)^(1/2) = 2x
(x^2+9)^(1/2) = 1/2x
x^2+9 = 1/4x^2

but that gives no solution.

Could someone point me to where I went wrong please?

 

[AlephZero]

Correct, the distance to row is sqrt(x^2+9) and the distance to walk is 5-x

You got the times wrong. Speed = distance / time, not distance * time.

 

[HallsofIvy]

"Let x be the distance from Q to the point on the shoreline at which the boat lands" is much more infomative than "let x be the distance from point Q"!