A fence is to be built to enclose a rectangular area of 310 square feet. The fence along three sides is to be made of material that costs 4 dollars per foot, and the material for the fourth side costs 12 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
To find the dimensions of the enclosure that is most economical to construct, we need to consider the costs of the fence along each side.
Let's suppose the length of the rectangular area is L feet and the width is W feet.
We are given that the fence along three sides (two widths and one length) costs $4 per foot, and the fence along the fourth side (the other length) costs $12 per foot.
The area of the rectangular area is given as 310 square feet, so we can write the equation:
L * W = 310
To find the dimensions that minimize the cost of the fence, we need to minimize the cost function. The cost function is the sum of the costs for each side of the fence.
Cost function, C = 2(W + L) * 4 + L * 12
= 8W + 8L + 12L
Now, we can substitute the value of W from the equation L * W = 310:
C = 8(310/L) + 8L + 12L
= 2480/L + 20L
To find the minimum value of the cost function, we need to find the critical points by taking the derivative with respect to L and setting it equal to zero:
dC/dL = -2480/L² + 20
= 0
Simplifying the equation:
-2480/L² + 20 = 0
Rearranging the equation:
2480/L² = 20
L² = 124
Taking the square root:
L = √124
≈ 11.1355
Since we cannot have a negative length, we take the positive value of L.
Now, substituting the value of L back into the equation L * W = 310:
11.1355 * W = 310
Solving for W:
W ≈ 27.84
So, the dimensions of the enclosure that is most economical to construct are approximately L = 11.1355 feet and W = 27.84 feet.