Question

A couple of newlyweds decides to s their house with a brand new white picket fence. They purchase 120 feet of fencing materials. If they build the fence in the shape of the rectangle that will maximize the area enclosed by the fence, what is the area that the fence will enclose? A. 900 ft^2B. 800 ft^2C. 500 ft^2 D. 0ft^2

Answer 1

Answer:

option (A) 900 ft²

Step-by-step explanation:

Let the length be L and width of the area be 'B'

now,

Perimeter of the fence = 2 (L + B)

also,

2 (L + B) = 120 feet

or

L + B = 60 ft

or

L = 60 - B ft

Now,

The area of the fencing ground, A = LB

or

A = (60 - B)B

A = 60B - B²

now,

differentiating the area with respect to width B, we get

$\frac{dA}{dB}= \frac{d(60B - B^2)}{dB}$

or

$\frac{dA}{dB}$ = 60 - 2B

for point of maxima or minima, put $\frac{dA}{dB}$ = 0

thus,

60 - 2B = 0

or

2B = 60

or

B = 30 ft

to check for maxima or minima

$\frac{d^2A}{dB^2}$ = - 2

since,

$\frac{d^2A}{dB^2}$ is negative, B = 30 ft is point of maxima

therefore,

L = 60 - B = 60 - 30 = 30 ft

Thus,

Maximum area of the fencing ground = 30 × 30 = 900 ft²

Hence,

The correct answer is option (A) 900 ft²