Question

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
the length of one side of the sheds base is □ ft.
the height of the shed is □ ft.

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.