Help with calculus homework?

Question

A small island is 2 miles from the nearest point P on the straight shoreline of a large lake. If Alice, who is on the island, can row 3 mph and walk 4 mph, how far down the shore from P should she land the boat in order to arrive at a town 10 miles down the shore from P in the least amount of time. Show your work.

Answer 1

Hi,

Let x = the place where she reaches shore to the right of p then 10 - x is the distance she walks the using the pythagorian theorem we have the distance she rows is sqrt(4 + x^2) now d = rt so t = d/r then sqrt(x^2 + 4) /3 + (10-x)/4 = total time.

Take the derivative giving us.

t' = 2x/(3*sqrt(x^2+4)) -1/4 = 0 so 8x -3sqrt(x^2 +4) = 0
then 8x = 3sqrt(x^2 +4) so squaring both sides gives us

64x^2 = 9x^2 + 36 then 55x^2 = 36 x^2 = 36/55 then x = 6/Sqrt(55) approx .8 mi then 10- x approx 9.2

Sound weird. Check my math.

Hope This Helps!