Answer:
L = 21.60 ft
W = 12.96 ft
Step-by-step explanation:
Let 'W' be the length of the more expensive side and 'L' be the length of the perpendicular sides. The expressions for the total area (A) and total cost (C) of the rectangular fence are:
[tex]A = WL=280\\C = 6*(2L+W)+14W=12L+20W[/tex]
Writing cost as a function of W:
[tex]L=\frac{280}{W}\\C =12*\frac{280}{W}+20W[/tex]
The cost will be minimal at the value of W for which the derivate of the cost function is zero:
[tex]C'=0 =-12*\frac{280}{W^2}+20\\W=\sqrt{\frac{3,360}{20}}\\W=12.96\ ft[/tex]
Therefore, the value of L is:
[tex]L = \frac{280}{W}= \frac{280}{12.96}\\L= 21.60\ ft[/tex]
Fencing of any rectangular region is basically represent perimeter of that rectangular region.
For most economical fencing, length L = 21.6 feet and width W = 12.96 feet.
Since, Area of rectangular region is 280 square feet.
And length is L and width is W.
Area of rectangular region = [tex]L * W = 280[/tex] square feet
[tex]W = 280/L[/tex]
Since, The fence along three sides is to be made of material that costs 6 dollars per foot, and the material for the fourth side costs 14 dollars per foot.
Therefore, Total cost , [tex]C = 6(2L+W)+14W = 12L + 20W[/tex]
Total cost , [tex]C = 12L + 5600/L[/tex]
To find most economical cost, differentiate above function and equate to zero.
Thus, [tex]C' = 12 - 5600/L^2\\ 12 - 5600/L^2 = 0\\ L^2 = 5600/12\\ L = 21.6 feet[/tex]
Substituting value of L in area equation.
We get, [tex]W = 12.96 feet[/tex]
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