A hardware store owner chooses to enclose an 800 square foot rectangular area in front of her store so that one of the sides of the store will be used as one of the four sides of the fence. If the two sides that come out from the store front cost $3 per running foot for materials and the side parallel to the store front costs $5 per running foot for materials, then find the dimensions of the fence that will minimize the cost to construct the fence. Round all dimensions to the nearest foot.

Question

A hardware store owner chooses to enclose an 800 square foot rectangular area in front of her store so that one of the sides of the store will be used as one of the four sides of the fence. If the two sides that come out from the store front cost $3 per running foot for materials and the side parallel to the store front costs $5 per running foot for materials, then find the dimensions of the fence that will minimize the cost to construct the fence. Round all dimensions to the nearest foot.

Answer 1

area=LW

L=800/W
Cost=3*2W+5L

cost=6W+4000/W

dcost/dw= 0=6-4000/W^2

W=sqrt(4000/6)

L= 800/W

Answer 2

Thank you, I got a very similar trying to do it the way our homework assignments were done. But your way seems much simpler. I got 25.8198898 by 30.76923077.

C=3x+3x+5y
=6x+5y

xy=800
y=800/x

c(x)=6x+5(800/x)
=6x+4000/x

c'(x)=6-4000/x^2

Find the critical value
6-4000/x^2=0
6=4000/x^2
6x^2=4000
x^2=4000/6
x^2=666.6666667
x=25.81988898

c"(x)=8000/x^3

C"(26)=8000/26^3
=.4551661356

xy=800
26y=800
y=800/26
y=30.76923077

The dimensions would by 26ft by 31ft.

Answer 3

To minimize the cost of constructing the fence, we need to find the dimensions that will result in the least amount of material used.

Let's assume that the side of the store used as one side of the fence is the length of the rectangle. So, if we let that side be x, then the width of the rectangle would be 800/x.

To determine the perimeter of the fence, we need to consider the four sides. One side is x, and three sides are 800/x.

Perimeter = x + 3 * (800/x)

The cost of the fence is determined by multiplying the perimeter by the cost per running foot for each side of the fence.

So, the cost function becomes:
Cost = cost per running foot * perimeter
Cost = (3 * x) + (5 * 3 * (800 / x))

To minimize the cost, we will take the derivative of the cost function with respect to x, set it equal to zero, and solve for x.

d(Cost)/dx = 3 - (5 * 3 * (800 / x^2))

Setting the derivative equal to zero:

3 - (5 * 3 * (800 / x^2)) = 0

Simplifying the equation:

3 - 12000 / x^2 = 0

Multiplying through by x^2:

3x^2 - 12000 = 0

Rearranging the equation:

3x^2 = 12000

Dividing through by 3:

x^2 = 4000

Taking the square root of both sides:

x = sqrt(4000) = 63.24

Since the dimensions are given in feet, we round the value of x to the nearest foot, which is 63.

Therefore, the dimensions of the fence that will minimize the cost to construct the fence are approximately 63 feet by 800/63 feet.