(1 point) A fence is to be built to enclose a rectangular area of 240 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 15 dollars per foot. Find the dimensions of the enclosure that is most economical to construct. Dimensions: x

Length = 19 ft

Width = 12.63 ft

Step-by-step explanation:

Let the length be 'L' and width be 'w'.

Given:

Area of the rectangular fence (A) = 240 ft²

Fencing cost for three sides = $5 per foot

Fencing cost for fourth side = $15 per foot

Area of the fence is given by the formula:

Now, Perimeter of first three sides =

Cost of fencing three sides = Cost per foot × Perimeter of 3 sides

Cost of fencing fourth side = Cost per foot × length of fourth side

Total cost of fencing,

Now, replace 'w' from equation (1). This gives,

Now, to minimize the total cost, the derivative of total cost with any of the dimensions must be 0.

Differentiating both sides with respect to length 'L', we get:

Therefore, the length is 19 feet.

Now, the width is determined using equation (1). So,

Therefore, the dimensions are length equal to 19 ft and width equal to 12.63 ft.

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