a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 729 cubic…

Explanation:

Step1: Define variables

Let the side - length of the square base be $x$ (in feet) and the height of the box be $h$ (in feet). The volume $V$ of the box is $V=x^{2}h$. Given $V = 729$, so $h=\frac{729}{x^{2}}$.

Step2: Calculate the cost function

The area of the base is $A_{base}=x^{2}$, and its cost is $3x^{2}$. The area of the roof is $A_{roof}=x^{2}$, and its cost is $6x^{2}$. The area of each of the four sides is $xh$, and the total area of the four sides is $4xh$, with a cost of $4.5\times4xh = 18xh$. The total cost function $C(x)$ is $C(x)=3x^{2}+6x^{2}+18xh=9x^{2}+18xh$. Substitute $h = \frac{729}{x^{2}}$ into the cost function: $C(x)=9x^{2}+18x\times\frac{729}{x^{2}}=9x^{2}+\frac{13122}{x}$.

Step3: Find the derivative of the cost function

Differentiate $C(x)$ with respect to $x$. Using the power - rule, if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. $C^\prime(x)=18x-\frac{13122}{x^{2}}$.

Step4: Set the derivative equal to zero

Set $C^\prime(x)=0$ to find the critical points. $18x-\frac{13122}{x^{2}} = 0$. Multiply through by $x^{2}$ to get $18x^{3}-13122 = 0$. Then $x^{3}=\frac{13122}{18}=729$, so $x = 9$.

Step5: Find the height

Substitute $x = 9$ into the equation for $h$: $h=\frac{729}{9^{2}}=\frac{729}{81}=9$.

Answer:

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