To find the dimensions of the enclosure that is most economical to construct, we need to minimize the cost of the fence.
Let's assume the length of the rectangular enclosure is "L" feet and the width is "W" feet.
The area of a rectangle is given by the formula A = length × width. In this case, the area is given as 270 square feet. So we have the equation:
L × W = 270
To minimize the cost of the fence, we need to find the dimensions that minimize the amount of expensive material used (the fourth side costing $15 per foot).
We can start by solving the equation for one of the variables in terms of the other. Let's solve it for L:
L = 270 ÷ W
Now, we can express the cost of the fence in terms of the width W. The total cost of the fence is the cost of the three sides (at $4 per foot) plus the cost of the fourth side (at $15 per foot). The cost is given by:
Cost = 3 × 4W + 15L
Substituting the expression for L from earlier, we get:
Cost = 3 × 4W + 15(270 ÷ W)
Simplifying this expression, we have:
Cost = 12W + 4050 ÷ W
To find the dimensions that minimize the cost, we need to find the value of W that minimizes the Cost equation.
One way to do this is by finding the derivative of the Cost equation with respect to W and setting it equal to zero. This will give us a critical point where the cost is minimized.
Taking the derivative of the Cost equation with respect to W, we have:
dCost/dW = 12 - 4050 ÷ W^2
Setting this derivative equal to zero and solving for W, we get:
12 - 4050 ÷ W^2 = 0
Simplifying, we find:
W^2 = 4050 ÷ 12
W^2 = 337.5
Taking the square root of both sides, we find:
W ≈ 18.36
Since W represents the width of the enclosure, we can round it to the nearest whole number to get 18 feet.
Now, using the equation L = 270 ÷ W, we can find the corresponding length:
L = 270 ÷ 18
L = 15
Therefore, the dimensions of the enclosure that is most economical to construct are approximately 18 feet by 15 feet.
By using these dimensions, we minimize the cost of the fence while enclosing an area of 270 square feet.