Question

A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $21 per foot and on the other three sides by a metal fence costing $7 per foot. If the area of the garden is 200 ft, find the dimensions of the garden minimizing the cost. (Let x be the length of the brick wall and y be the length of an adjacent side in feet.)

Answer 1

Answer:

L = 6.33 foot

W = 31.62 foot

Step-by-step explanation:

costing of one side is $21 per foot and  the costing of other three sides is $ 7 per foot.

Area of rectangular garden, A = 200 ft²

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

A = L x W

200 = L x W .... (1)

Total cost of fencing is

C = 21 x L + 7 ( 2L + W)

C = 35 L + 7 W .... (2)

Substitute the value of W from equation (1) in equation (2)

$C = 35 L + 7\times \frac{200}{L}$

$C = 35 L + \frac{1400}{L}$

Differentiate with respect to L on both the sides:

$\frac{dC}{dL} = 35 - \frac{1400}{L^{2}}$

Put it equal to zero for maxima and minima

$35 - \frac{1400}{L^{2}}=0$

L = 6.33 foot

W = 31.62 foot

So, the cost of fencing is minimum when L = 6.33 foot , W = 31.62 foot