Question

A rectangular study area is to be enclosed by a fence and divided into two equal parts, with the fence running along the division parallel to one of the sides. If the total area is 384 square feet, find the dimensions of the study area that will minimize the total length of the fence. How much fencing will be required?

Answer 1

Answer:

Explanation:

Let l be the length of the study area and w be the width of study area.

Area of rectangular part = l x w

A = l x w

384 = l x w

w = 384 / l .... (1)

length of the fence is equal to the perimeter.

P = 2 ( l + w)

$P = 2\left ( l +\frac{384}{l} \right )$   from equation (1)

Differentiate with respect to l

$\frac{dP}{dl} = 2\left ( 1 -\frac{384}{l^{2}} \right )$

for maxima and minima, it is equal to zero

$\left ( 1 -\frac{384}{l^{2}} \right )=0$

l = 19.6 ft

so w = 19.6 ft

So, the required fencing = 2 ( 19.6 + 19.6) = 78.4 ft