A fence is to be built to enclose a rectangular area of 210 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 14 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
A fence is to be built to enclose a rectangular area of 210 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 16 dollars per foot.
To find the dimensions of the enclosure that is most economical to construct, we can use the concept of calculus optimization.
Let's assume the length of the rectangular area is L (in feet) and the width is W (in feet).
The area of the rectangular area is given as 210 square feet, so we have the equation:
L * W = 210
We need to minimize the cost of the fence while keeping the area constant.
The cost of the fence along three sides is 3 dollars per foot, so the cost of those sides is 3 * (L + W + L) = 6L + 3W dollars.
The length of the fourth side is W, and its cost is 14 dollars per foot, so the cost of the fourth side is 14W dollars.
The total cost of the fence is:
Total Cost = 6L + 3W + 14W
Now, we can express the total cost in terms of a single variable by using the equation for the area:
W = 210 / L
Substituting this value of W into the total cost equation, we have:
Total Cost = 6L + 3(210 / L) + 14(210 / L)
Simplifying this equation, we get:
Total Cost = 6L + 630 / L + 2940 / L
Total Cost = (6L^2 + 630 + 2940) / L
To find the dimensions that minimize the cost, we need to find the critical points by taking the derivative of the Total Cost equation with respect to L and setting it equal to zero.
d(Total Cost) / dL = (12L - 2940) / L^2
Setting this equal to zero, we have:
12L - 2940 = 0
12L = 2940
L = 245 feet
Substituting this value of L back into the equation for W:
W = 210 / L
W = 210 / 245
W = 0.857 feet
Therefore, the dimensions of the enclosure that are most economical to construct are approximately 245 feet (length) and 0.857 feet (width).