a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 1458 cubic…

a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 1458 cubic feet. the concrete for the base costs $4 per square foot, the material for the roof costs $2 per square foot, and the material for the sides costs $1.50 per square foot. find the dimensions of the most economical shed\nthe length of one side of the sheds base is □ ft.\nthe height of the shed is □ ft.

a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 1458 cubic feet. the concrete for the base costs $4 per square foot, the material for the roof costs $2 per square foot, and the material for the sides costs $1.50 per square foot. find the dimensions of the most economical shed\nthe length of one side of the sheds base is □ ft.\nthe height of the shed is □ ft.

Answer

Explanation:

Step1: Define variables and volume formula

Let the side length of the square base be (x) (in feet) and the height be (h) (in feet). The volume (V=x^{2}h), and since (V = 1458), we have (h=\frac{1458}{x^{2}}).

Step2: Write the cost function

The area of the base is (A_{base}=x^{2}), cost of the base (C_{base}=4x^{2}). The area of the roof is (A_{roof}=x^{2}), cost of the roof (C_{roof}=2x^{2}). The area of the four - sides is (A_{sides}=4xh), cost of the sides (C_{sides}=1.5\times4xh = 6xh). The total cost function (C(x)=4x^{2}+2x^{2}+6x\times\frac{1458}{x^{2}}). Simplify (C(x)=6x^{2}+\frac{8748}{x}).

Step3: Find the derivative of the cost function

Using the power rule, if (y = ax^{n}), then (y^\prime=anx^{n - 1}). (C^\prime(x)=12x-\frac{8748}{x^{2}}).

Step4: Set the derivative equal to zero and solve for (x)

(12x-\frac{8748}{x^{2}} = 0). Multiply through by (x^{2}) to get (12x^{3}-8748 = 0). (x^{3}=\frac{8748}{12}=729). Take the cube - root: (x = 9).

Step5: Find the value of (h)

Since (h=\frac{1458}{x^{2}}), substitute (x = 9) into the formula. (h=\frac{1458}{9^{2}}=\frac{1458}{81}=18).

Answer:

The length of one side of the shed's base is (9) ft. The height of the shed is (18) ft.