A fence is to be built to enclose a rectangular area of 800 square feet. The fence along 3 sides is to be made of material $4 per foot. The material for the fourth side costs $12 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built.
if the expensive side is x and the other dimension is y, then the cost c is
c = 4(x+2y) + 12x
But, we know the area is xy=800, so y = 800/x and the cost is now
c = 4(x+1600/x) + 12x
minimum cost when dc/dx=0, so we need
dc/dx = -16(400-x^2)/x^2
dc/dx=0 when x=20, so the fence is 20x40
Don't have an answer. Just need to know how the calculations are done.
. Afence is to be built to enclose a rectangular area of 800 square feet. The fence along three sides is to be made of material that costs Birr2 per foot. The material for the fourth side costs Birr6 per foot. Find the dimension of the rectangular that will allow the most economical fence to be built.
To find the dimensions of the rectangle that will allow for the most economical fence to be built, we need to consider the cost of the fence for different dimensions. Let's assume the length of the rectangle is 'x' feet and the width is 'y' feet.
The area of the rectangle is given as 800 square feet, so we have the equation:
x * y = 800
We also know that three sides of the fence will be made of material costing $4 per foot, and the remaining side will cost $12 per foot. Therefore, the total cost of the fence is given by:
C(x, y) = 3(4x) + 12y = 12x + 12y
We can rewrite the area equation as:
y = 800/x
Substituting this into the cost equation, we have:
C(x) = 12x + 12(800/x)
To find the dimensions that minimize the cost, we differentiate this equation with respect to x and set it equal to 0:
dC/dx = 12 - 12(800/x^2) = 0
Simplifying, we have:
12 - (9600/x^2) = 0
Rearranging, we get:
12 = 9600/x^2
x^2 = 9600/12
x^2 = 800
Taking the square root of both sides, we get:
x = sqrt(800) ≈ 28.28 ft
Substituting this value back into the equation for y, we have:
y = 800/x ≈ 800/28.28 ≈ 28.28 ft
Therefore, the dimensions that will allow for the most economical fence to be built are approximately 28.28 ft by 28.28 ft.