Question

Mr. Aydlett is on an island 2 miles offshore and wishes to reach a coastal village 6 miles down a straight shoreline from the point on shore nearest the island. He can row his boat 2 mph and can walk 5 mph. Where should he land his boat to reach the village in the least amount of time? What is the minimum time?

Answer 1

Answer:t=2.11 hr

Step-by-step explanation:

Given

Aydlett is 2 miles offshore and village is 6 miles down a straight line from the Point on the shore nearest the island

Person can row boat at 2 mph in water and can walk 5 mph in land

Let us suppose Person land the boat at a x miles  from Point on the shore

thus time taken by him to reach

$t_1=\frac{\sqrt{x^2+2^2}}{2}$

Time taken person to reach village by land is

$t_2=\frac{6-x}{5}$

total time$=\frac{\sqrt{x^2+2^2}}{2}+\frac{6-x}{5}$

we need the time to be least so differentiate t w.r.t to x

$\frac{\mathrm{d} t}{\mathrm{d} x}=\frac{2x}{2\cdot 2\cdot \sqrt{4+x^2}}-\frac{1}{5}$

Equating Above term to zero to get minimum time

$\frac{x}{2\sqrt{4+x^2}}=\frac{1}{5}$

$25x^2=16+4x^2$

$x=\frac{4}{\sqrt{21}}$

Substituting x in time equation

$t=\frac{\sqrt{4+0.761}}{2}+\frac{6-0.872}{5}$

$t=2.116 hr$