Answer:
Pine board side = 16.4 ft
Steel fencing side = 57.5 ft
Step-by-step explanation:
Let 'B' be the length of each side constructed of pine boards, and 'S' be the length of the side with the steel fencing, the area (A) and cost (C) functions are:
[tex]945 = B*S\\B=\frac{945}{S} \\C= 4*S+7*2B\\C=4S+\frac{13,230}{S}[/tex]
The value of S for which the derivate of the cost function is zero, minimizes cost:
[tex]C'=0=4+\frac{-13,230}{S^2}\\ S=\sqrt{3,307.5} \\S=57.5 ft[/tex]
The value of B is:
[tex]B=\frac{945}{57.5}\\B=16.4 \ ft[/tex]
Pine board side = 16.4 ft
Steel fencing side = 57.5 ft
The dimensions that minimize the enclosure is 57.6 ft by 16.4 ft
Let the dimension of the store be L by W.
So, the area is:
Area = LW
The area is given as 945.
So, we have:
[tex]LW = 945[/tex]
Make L the subject
[tex]L = \frac{945}W[/tex]
The perimeter of the store is:
[tex]P = L + 2W[/tex] ---because one side will be covered by an external wall
The pine board fencing costs $7/running foot and the steel fencing costs $4/running foot.
So, the cost function is:
[tex]C = 4L + 7 * 2W[/tex]
This gives
[tex]C = 4L + 14W[/tex]
Substitute [tex]L = \frac{945}W[/tex]
[tex]C = 4*\frac{945}W + 14W[/tex]
[tex]C = \frac{3780}W + 14W[/tex]
Differentiate
[tex]C' = -3780W^{-2} + 14[/tex]
Set to 0
[tex]-3780W^{-2} + 14 = 0[/tex]
Divide through by 14
[tex]-270W^{-2} + 1 = 0[/tex]
Collect like terms
[tex]-270W^{-2} =- 1[/tex]
Multiply both sides by -W^2
[tex]W^2 = 270[/tex]
Take the square root of both sides
[tex]W = 16.4[/tex]
Recall that:
[tex]L = \frac{945}W[/tex]
So, we have:
[tex]L = \frac{945}{16.4}[/tex]
[tex]L = 57.6[/tex]
Hence, the dimensions that minimize the enclosure is 57.6 ft by 16.4 ft
Read more about areas at:
https://brainly.com/question/16997207
