A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To pr

Question

A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for the herd, the pasture must contain 231, 200 square meters. No fencing is required along the river. What dimensions will use the least amount of fencing? (See figure below.) x = m y = m You are in a boat 2 miles from the nearest point of the coast. You are to go to point Q, located 3 miles down the coast and 1 mile inland (see figure below). You can row at a rate of 2 miles per hour and you can walk at a rate of miles per hour. Toward what point on the coast should you row in order to reach point Q in the least amount of time? x = mile

Explanation / Answer

area = x*y
231200 =x*y
y = 231200/x

amount of fence required,
F = 2y+x
= 2*(231200/x) + x
= 462400/x + x

F(x) = 462400/x + x
F'(x) = -462400/x^2 + 1

put F'(x) = 0
-462400/x^2 + 1 = 0
x^2 = 462400
x = 680

for x<680,F'(x) is negative
for x>680,F'(x) is positive
so,
x = 680 minimizes fence required

y = 231200/680 = 340

Answer:
x = 680 m
y = 340 m

I am allowed to answer only 1 question at a time

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