The cost of the least expensive fence is $3200
Step-by-step explanation:
The given is:
We need to find the cost of the least expensive fence
Assume that the length of each side opposite to North or South is x feet and the length of each other sides is y feet
∵ The length of the rectangle = x feet
∵ The width of the rectangle = y feet
∵ The rectangular area is 20,000 ft²
- Area of a rectangle = length × width
∴ x × y = 20,000
- Divide both sides by x to find y in terms of x
∴ [tex]y=\frac{20,000}{x}[/tex]
The fence's length is equal to the perimeter of the rectangular area
∵ Perimeter of the rectangle = 2 length + 2 width
∴ Perimeter of the rectangle = 2x + 2y
∵ Fencing material costs $4 per foot for the two sides facing
North and South
∴ x costs $4 per foot
∵ The cost of the other two sides is $8 per foot
∴ y costs $8 per feet
The cost of the fence is the sum of the products of 4 , 2x and 8 , 2y
∵ The cost of the fence (C) = 4(2x) + 8(2y)
∴ C = 8x + 16y
- Substitute y by its value above
∴ [tex]C=8x+16(\frac{20,000}{x})[/tex]
∴ [tex]C=8x+\frac{320,000}{x}[/tex]
To find the least expensive differentiate C with respect to x and equate the answer by 0 to find the value of x
∵ [tex]\frac{320,000}{x}[/tex] can be written as [tex]320,000x^{-1}[/tex]
∴ [tex]C=8x+320,000x^{-1}[/tex]
∵ [tex]\frac{dC}{dx}=8(1)x^{1-1}+320,000(-1)x^{-1-1}[/tex]
∴ [tex]\frac{dC}{dx}=8-320,000x^{-2}[/tex]
- Equate [tex]\frac{dC}{dx}[/tex] by zero
∴ [tex]8-320,000x^{-2}=0[/tex]
∵ [tex]-320,000x^{-2}=-\frac{320,000}{x^{2}}[/tex]
∴ [tex]8-\frac{320,000}{x^{2}}=0[/tex]
- Subtract 8 from both sides
∴ [tex]-\frac{320,000}{x^{2}}=-8[/tex]
- Multiply both sides by x²
∴ - 320,000 = - 8x²
- Divide both sides by -8
∴ 40,000 = x²
- Take √ for both sides
∴ 200 = x
Substitute x in the equation of C to find the least cost of fence
∵ [tex]C=8(200)+\frac{320,000}{(200)}[/tex]
∴ C = 1600 + 1600
∴ C = 3200
The cost of the least expensive fence is $3200
Learn more:
You can learn more about the differentiation in brainly.com/question/4279146
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