A small island is 6 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?
The boat should be landed ____ miles down the shore from P.
(Please give exact answer (Ex. 15/square root 6)
The boat should be landed ____ miles down the shore from P.
(Please give exact answer (Ex. 15/square root 6)
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Let x be the distance down the shore where the boat is landed.
t1 = time to row to the shore sqrt(6^2 + x^2)/3
t2 = time to walk from the point where the boat is landed = (10 - x)/4
t = total time = t1 + t2 = sqrt(6^2 + x^2)/3 + (10 - x)/4
t = sqrt(36 + x^2)/3 + 5/2 - x/4
dt/dx = x/[3*sqrt(36 + x^2)] - 1/4 = 0
x^2/(36 + x^2) = 9/16
16*x^2 = 9*(36 + x^2)
7*x^2 = 9*36
x = 18/sqrt(7) = (18/7)*sqrt(7)
So the distance is (18/7)*sqrt(7)
t1 = time to row to the shore sqrt(6^2 + x^2)/3
t2 = time to walk from the point where the boat is landed = (10 - x)/4
t = total time = t1 + t2 = sqrt(6^2 + x^2)/3 + (10 - x)/4
t = sqrt(36 + x^2)/3 + 5/2 - x/4
dt/dx = x/[3*sqrt(36 + x^2)] - 1/4 = 0
x^2/(36 + x^2) = 9/16
16*x^2 = 9*(36 + x^2)
7*x^2 = 9*36
x = 18/sqrt(7) = (18/7)*sqrt(7)
So the distance is (18/7)*sqrt(7)