a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 125 cubic…

Explanation:

Step1: Recall volume formula

The volume $V$ of a box with square - base of side length $x$ and height $h$ is $V=x^{2}h$. Given $V = 125$, so $h=\frac{125}{x^{2}}$.

Step2: Calculate area and cost of each part

• Base area $A_{base}=x^{2}$, cost of base $C_{base}=4x^{2}$.
• Roof area $A_{roof}=x^{2}$, cost of roof $C_{roof}=9x^{2}$.
• Side - area: Each side has area $xh$, and there are 4 sides. So $A_{sides}=4xh$, and cost of sides $C_{sides}=6.5\times4xh = 26xh$.
• Total cost $C = C_{base}+C_{roof}+C_{sides}=4x^{2}+9x^{2}+26xh$. Substitute $h=\frac{125}{x^{2}}$ into the cost formula: $C = 13x^{2}+\frac{3250}{x}$.

Step3: Find derivative of cost function

Differentiate $C(x)=13x^{2}+\frac{3250}{x}=13x^{2}+3250x^{-1}$ with respect to $x$. Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $C^\prime(x)=26x-3250x^{-2}=26x-\frac{3250}{x^{2}}$.

Step4: Set derivative equal to zero

Set $C^\prime(x) = 0$: [ \begin{align*} 26x-\frac{3250}{x^{2}}&=0\ 26x&=\frac{3250}{x^{2}}\ 26x^{3}&=3250\ x^{3}& = 125\ x&=5 \end{align*} ]

Step5: Find the height

Substitute $x = 5$ into the height formula $h=\frac{125}{x^{2}}$. Then $h=\frac{125}{25}=5$.

Answer:

The length of one side of the shed's base is $5$ ft. The height of the shed is $5$ ft.