A small island is 4 miles from the nearest point P on the straight shoreline of

Question

A small island is 4 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the boat be landed in order to arrive at a town 7 miles down the shore from P in the least time? Let be the distance (in miles) between point P and where the boat lands on the lakeshore. (a) Enter a function that describes the total amount of time the trip takes as a function of the distance . (include units) (b) What is the distance that minimizes the travel time? (include units) (c) What is the least travel time? The least travel time is (include units)

Explanation / Answer

A small island is 4 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 10 miles down the shore from P in the least time? Let be the distance (in miles) between point P and where the boat lands on the lakeshore. The island, P and the town forms a right triangle. Let X be the point between P and the town where the boat lands, and x be the distance between X and P. The distance Db traveled by boat = sqrt( 4^2 + x^2 ) The distance Dw traveled by foot = 10 - x t = travel time = tb + tw = time spent by boat + time spent walking tb = Db / 3 tw = Dw / 4 t = (1/3) sqrt(16 + x^2) + (1/4) (10 - x) travel time minimized when dt/dx = 0; solve for x then solve for {Db,Dw,tb,tw,t}

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