Question

A homeowner wants to build a fence to enclose a 180 square yard rectangular area in his backyard. Along one side the fence is to be made of heavy-duty material costing $9 per yard, while the material along the remaining three sides costs $1 per yard. Determine the least cost to the homeowner.

Answer 1

Answer:$ 366.71

Step-by-step explanation:

Given

Area of rectangular area $A=180 yards^2$

Let Length and Breadth be L and b yard

at one side it cost $ 9 per yard and $ 1 per yard

Area =Lb

Total cost C=9L+b

substitute the value of L i.e. $L=\frac{180}{b}$

$C=9\times \frac{180}{b}+b$

differentiate C w.r.t to b

$\frac{\mathrm{d} C}{\mathrm{d} b}=-\frac{9\times 180}{b^2}+1$

Put $\frac{\mathrm{d} C}{\mathrm{d} b}=0$  for max/min value

$b^2=9\times 180$

$b=\sqrt{9\times 180}$

$b=40.24 yd$

therefore Length $L=\frac{180}{20.24}=4.47 yd$

Total Cost $C=9\times 40.24+4.47 =\ 366.71$