Question

A man plans to make a rectangular garden with one side adjoining a neighbor’s yard. The garden is to be 675 ft2. If the neighbor agrees to pay half of the dividing fence, what should the dimensions of the garden be to ensure a minimum cost for the man?

Answer 1

Answer:

L = W = 26 ft.

Explanation:

Let the length of the garden is L and the width is W.

area of garden , A = L x W

675 = L x W ... (1)

Costing is minimum when the perimeter is minimum.

Perimeter, P = 2 ( L + W)

$P=2\left ( L+\frac{675}{L} \right )$

For maxima and minima, differentiate perimeter with respect to L.

$\frac{dP}{dL}=2\left ( 1-\frac{675}{L^{2}} \right )$

It should be zero for maxima and minima

L² = 675

L = 26 ft

W = 675/26 = 26 ft

Now, $\frac{d^{2}P}{dL^{2}}=4\frac{675}{L^{3}}$

It is positive, so the costing is minimum.

So, length and width of the garden is 26 ft.