Boat Shoreline Question

Question

Jane is c=3 mi offshore in a boat and wishes to reach a coastal village b=8 mi down a straight shoreline from the point nearest the boat. She can row 2 mph and walk 5 mph. At what point x miles away should she land her boat to reach the village the quickest?

Answer 1

Make a sketch:

Since we are being asked "to get there quickest", that means we want to minimize time. So we need to develop a function for the time it takes, and minimize it, i.e., take the derivative, set it equal to zero, and find the minimum time.

time = distance/rate

The distance to row is found using the Pythagorean Theorem:

d_row = √(3² + x²) = √(x² + 9)

The rate for rowing is 2 so the time spent rowing is √(x² + 9)/2 or (1/2)(x² + 9)^1/2

The distance to walk is 8 - x.

The rate for walking is 5 so the time spent walking is (8 - x)/5 = 8/5 - (1/5)x

So the function for time is:

t(x) = (1/2)(x² + 9)^1/2 + 8/5 - (1/5)x

Take the derivative:

t'(x) = (1/4)(x² + 9)^-1/2(2x) - 1/5

Then set it equal to zero and solve:

(1/4)(x² + 9)^-1/2(2x) - 1/5 = 0

(1/4)(x² + 9)^-1/2(2x) = 1/5

1/(x² + 9)^1/2 = 2/(5x)

cross multiply to get:

2(x² + 9)^1/2 = 5x

[(x² + 9)^1/2]² = (2.5x)²

x² + 9 = 6.25x²

9 = 5.25x²

x² = 9/5.25

x = 1.309 miles

Answer 2

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