Answer:
The length that would produce the least expenses is [tex]A= 300 ft[/tex]
The width that would produce the least expenses is [tex]B= 150 ft[/tex]
Step-by-step explanation:
From the question we are told that the
Required area to enclose is [tex]A = 45,000 ft^2[/tex]
Fencing material cost is [tex]C =[/tex]$4 per foot for north and south
Fencing material cost for east and west [tex]C_{E/W} =[/tex] $8
The diagram for this question is shown on the first uploaded image
From the diagram is mathematically evaluated as
AB = 45000
=> [tex]B = \frac{45000}{A}[/tex]
The overall cost of building this fence is
[tex]T = 2 A (4) + 2(B)(8)[/tex]
Substituting for B in the equation above
[tex]T = 8A +16 (\frac{45000}{A} )[/tex]
differentiating both sides with respect to x
[tex]T' (A) = 8 - \frac{16 *45000}{A^2}[/tex]
At minimum possible cost [tex]T'(A) = 0[/tex]
=> [tex]8 - \frac{16 *45000}{A^2} = 0[/tex]
[tex]8A^2 = 16*45000[/tex]
[tex]A^2 = \frac{16*45000}{8}[/tex]
[tex]A = \sqrt{\frac{ 16*45000}{8} }[/tex]
[tex]= 300 ft[/tex]
Then B is mathematically evaluate as
[tex]B = \frac{45000}{300}[/tex]
[tex]= 150 ft[/tex]
Then the maximum is mathematically evaluated as
[tex]T = 8 (300) + 16(150)[/tex]
=$4800
