Question

fence is to be built to enclose a rectangular area of 250 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 13 dollars per foot. Find the dimensions of the enclosure that is most economical to construct

Answer 1

Answer:

23.05 x 10.85

Step-by-step explanation:

Area of Rectangular Fence = L * W = 250$ft^{2}$

L * W = 250

W = $\frac{250}{L}$                                      ................................... (i)

Perimeter = 2L + 2W

Three sides cost 4 dollars per foot = 4(2L) + 4W

(The two lengths must be constructed with the 4 dollars materials)

Fourth side costs 13 dollars per foot = 13W

Cost = 4(2L) + 4W + 13W

C = 8L + 17W                         ...................................... (ii)

Substitute (i) into (ii)

C = 8L + 17 ($\frac{250}{L}$)

C = 8L + $\frac{4250}{L}$  

Plotting this equation with L on the x-axis and cost on the y-axis

minimum cost will be when L ≈ 23.05 ft

W = $\frac{250}{23.05}$ ≈ 10.85 ft

minimum cost = 8(23.05) + $\frac{4250}{23.05}$

minimum cost = 184.4 + 184.382

minimum cost ≈ 368.78 dollars

Answer 2

The dimensions of the enclosure that is most economical to construct is L = 23.05 ft and W = 10.85 ft and this can be determined by using the formula of area and perimeter of a rectangle.

Given :

• The fence is to be built to enclose a rectangular area of 250 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 13 dollars per foot.

Given that the area of a rectangle is given by the formula :

$\rm L\times W = 250$

$\rm W =\dfrac{250}{L}$    ---- (1)

Now, the perimeter of a rectangle is given by:

$\rm P = 2(L+W)$

Given that the fence along three sides is to be made of material that costs 4 dollars per foot = 4(2L) + 4W

Also given that the fourth side costs 13 dollars per foot = 13W

Now, the total cost is given by:

$\rm Total\; Cost = 8L + 4W+13W$

$\rm Total\; Cost = 8L + 17W$    --- (2)

Now, substitute the value of W in the above equation.

$\rm Cost = 8L +17\times \dfrac{250}{L}$

Now, for minimum cost differentiate the above equation with respect to L.

$\rm C' = 8 -\dfrac{4250}{L^2}$

Now, equate the above equation to zero.

$\rm L^2 = \dfrac{4250}{8}$

L = 23.05 ft.

Now, put the value of L in equation (1).

$\rm W = \dfrac{250}{23.05}$

W = 10.85 ft.

So, the dimensions of the enclosure that is most economical to construct is L = 23.05 ft and W = 10.85 ft.

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https://brainly.com/question/1631786