Question

Apel Sinoman has 100 ft of fencing material to enclose a rectangular exercise run for her dog. One side of the run will border her​ house, so she will only need to fence three sides. What dimensions will give the enclosure the maximum​ area?

Answer 1

Answer:

For maximum area of the rectangular exercise run dimensions will be 50ft by 25ft.

Step-by-step explanation:

Let the length of the rectangular exercise run = l ft

and width of the run = w ft

Sinoman has to cover a rectangular exercise run from three sides with the fencing material,

So length of the material = (l + 2w) ft

l + 2w = 100

l = 100 - 2w --------(1)

Area of the rectangular area covered = Length × width

A = lw

A = w(100 - 2w)    [(l = 100 - 2w)from equation (1)

For maximum area we find the derivative of area and equate it to zero.

$\frac{dA}{dw}=\frac{d}{dw}[w(100-2w)]$

$A'=\frac{d}{dw}(100w-2w^{2} )$

A' = 100 - 4w

For A' = 0

100 - 4w = 0

4w = 100

w = 25 ft

From equation (1)

l = 100 - 2w

l = 100 - 2×(25)

l = 50 ft

Therefore, for maximum area of the rectangular exercise run dimensions will be 50ft by 25ft.