a rectangular fence is constructed that will enclose 100 ft sq of land. three sides will be built from fencing that costs 10 dollars per foot and the 4th side will be built from fencing that costs 23 dollars per foot. find the dimensions of the rectangular fence that minimizes the cost of materials
x by y
x y = 100
23 x + 10 x + 20 y = cost = c
33 x + 20 y = c
y = 100/x
so
33x + 2000/x = c
dc/dx = 0 for min = 33 -2000/x^2
33 x^2 = 2000
x^2 = 60.6
x = 7.785 ft
y = 12.845 ft
To minimize the cost of materials, we need to find the dimensions of the rectangular fence that will still enclose 100 sq ft of land.
Let's assume the length of the rectangular fence is "l" and the width is "w".
The equation for the area of a rectangle is A = l * w, and in our case, the area is 100 sq ft. So we have the equation:
l * w = 100
We also know that three sides of the fence will be built using fencing that costs $10 per foot, and the fourth side will use fencing costing $23 per foot.
The cost of the three sides is given by:
cost_of_three_sides = 10 * 2l + 10 * w
The cost of the fourth side is given by:
cost_of_fourth_side = 23 * l
The total cost is the sum of the cost of three sides and the cost of the fourth side:
Total Cost = cost_of_three_sides + cost_of_fourth_side
To minimize the cost, we need to find the dimensions that minimize the Total Cost. We can do this by substituting the equations for cost_of_three_sides and cost_of_fourth_side into the Total Cost equation:
Total Cost = (10 * 2l + 10 * w) + (23 * l)
Simplifying this equation, we get:
Total Cost = 20l + 10w + 23l
Total Cost = 43l + 10w
Now we can substitute the value of w from the area equation into the Total Cost equation:
Total Cost = 43l + 10(100/l)
To minimize the cost, we can take the derivative of the Total Cost equation with respect to "l" and set it equal to zero, then solve for "l". The value of "l" will give us the length of the fence that minimizes the cost.
Let's find the derivative of the Total Cost equation:
d(Total Cost) / dl = 43 - 1000/l^2
Setting this derivative equal to zero:
43 - 1000/l^2 = 0
Solving for "l":
1000/l^2 = 43
l^2 = 1000/43
l^2 ≈ 23.26
Taking the square root to solve for l:
l ≈ 4.82
Substituting this value of "l" back into the area equation:
l * w = 100
4.82 * w = 100
w ≈ 20.74
So, the dimensions of the rectangular fence that minimizes the cost of materials are approximately 4.82 ft by 20.74 ft.