A small island is 5 miles from the nearest point P on the straight shoreline of

Question

A small island is 5 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the boat be landed in order to arrive at a town 8 miles down the shore from P in the least time? Let x be the distance (in miles) between point P and where the boat lands on the lakeshore. Enter a function T(x) that describes the total amount of time the trip takes as a function of the distance x. T(x) =

Explanation / Answer

The trip consists of two parts. The rowing part is the hypotenuse of right angled triangle
whose sides are the distance from P to the island, which is 5, and the distance between P and
the landing point of the rowboat on the shore, which is x

so this part of trip is sqrt(25+ x^2)

The 2nd part is the walking part, which is (8-x)

Distance = rate times time (D = rt), so to get the time you have t = D/r. We must divide each
of the trip by the appropriate rate to get the time.

a) T(x) = sqrt(25+x^2)/3 + (8-x)/4

To find minimum time required take derivative of the T(x) function and find it's zeros

T'(x) = x/(3(sqrt(25+x^2)) - 1/4 = 0

x/(3(sqrt(25+x^2)) = 1/4
4x = 3sqrt(25+x^2
16x^2 = 9(25+x^2) = 225 + 9x^2
7x^2 = 225
x^2 = 225/7
x = sqrt(225/7) = 5.669467 miles

T(x) = 3.602386382 hours

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