Answer:
length = 60 foot, width = 30 foot
Step-by-step explanation:
Area of rectangular part, A = 1800 ft²
Cost of fencing three sides is $ 6 per foot and cost of one side fencing is $18 per foot.
Let the length of the rectangle is l and the width of the rectangle is W.
Area = Length x width
A = L x W
1800 = L x W ...... (1)
Total cost of fencing, C = 6 x ( L + W + L) + 18 x W
C = 6 (2L + W) + 18 W
C = 12 L + 24 W
Substitute the value of W from equation (1), [tex]W=\frac{1800}{L}[/tex] in equation (2)
[tex]C=12L+24\times \frac{1800}{L}[/tex]
[tex]C=12L+\frac{43200}{L}[/tex]
Differentiate both sides with respect to L:
[tex]\frac{dC}{dL}=12-\frac{43200}{L^{2}}[/tex]
Put it equal to zero for maxima and minima
[tex]12-\frac{43200}{L^{2}}=0[/tex]
L = 60 foot
and W = 30 foot
So, the costing is minimum for length = 60 foot and the width = 30 foot.
