Question

A hunter is at a point on a riverbank. He wants to get to his cabin located 2 Miles North and 8 Miles West. He can travel 5 Mph on the riverbank but only 3 Mph on The Rough Rocky Ground. How far Upriver should he go In order to reach the cabin in the minimum amount of time? Choose The Correct Number Of Miles The Hunter Should Travel Along The Riverbank
(A) 6.50
(B) 2.20.
(C) 5.80.
(D) 1.50.

Answer 1

Answer:

Option A

Step-by-step explanation:

A hunter travels to his cabinet located at 2 Miles North and 8 Miles West.

Let "d" be the distance traveled before reaching the rocky ground at point say "P". Let the cabin be at "C"

The distance between C and P is given by

$\sqrt{(2^2+(8-d)^2}$

Let "t" be the time taken to travel distance CP.

$T = \frac{d}{5} + \frac{\sqrt{2^2+(8-d)^2} }{3}$

Minimum time will be when "t" is equal to zero

Differentiating the above equation wrt "d", we get -

$= \frac{1}{5} + \frac{1}{3} (\frac{-2(8-d)}{2\sqrt{2^2+(8-d)^2} } )= 0\\\frac{1}{5} = \frac{1}{3} \frac{(8-d)}{\sqrt{4+(8-d)^2} } \\3\sqrt{4+64+d^2-16d} = 40-5d$

On squaring both the sides , we get -

$3\sqrt{4+64+d^2-16d} = 40-5d\\9(68+d^2-16d)= 1600+25d^2 -400d\\612+9d^2-144d=1600+25d^2 -400d\\16d^2-256d+988=0\\d_1 = 9.5\\d_2 = 6.5$

"d" cannot be greater than 8 miles.

Hence, the value of "d" is equal to $6.5$

Option A