Question

A sherman is in a rowboat on a lake and 3
km from shore. He wishes to reach a store 2
km down the (straight) shore. He can row at
5 km/h and run at 13 km/h. To what point
down-shore should he row to get to the store
as quickly as possible?
8. The cross-section of a tunnel

Answer 1

Let his position in the lake be b, let A be the point on shore closest to him, let S be the store
so AS = 2, AB = 3
Let P be the point between A and S.
So triangle ABP is right angled and
BP^2 = x^2 + 9
BP = (x^2+9)^(1/2)

recall that Time = Distance/Rate

Total Time = T
= ((1/5)(x^2+9)^(1/2) + (2-x)/13
dT/dx = (1/10)(x^2+9)^(-1/2) (2x) - 1/13
= 0 for a min of T

x/5√(x^2 + 9) = 1/13
5√(x^2 + 9) = 13x
square both sides
25(x^2 + 9) = 169x^2
25x^2 + 225 = 169x^2
144x^2 = 225
12x = 15
x = 15/12

so distance to store = 2 - x
= 2 - 15/12 = 3/4

He should row to a point 3/4 km from the store
(or 1 1/4 km from the point A)