a rectangular tank that is 108 ft³ with a square base and open top is to be constructed of sheet steel of a…

Explanation:

Step1: Define variables

Let the side - length of the square base be $x$ (in feet) and the height of the tank be $h$ (in feet). The volume $V$ of the rectangular tank is given by $V=x^{2}h$. Since $V = 108$, we have $h=\frac{108}{x^{2}}$.

Step2: Find the surface - area formula

The surface area $S$ of the open - top tank (with a square base) is $S=x^{2}+4xh$. Substitute $h=\frac{108}{x^{2}}$ into the surface - area formula: $S=x^{2}+4x\cdot\frac{108}{x^{2}}=x^{2}+\frac{432}{x},x>0$.

Step3: Differentiate the surface - area function

Differentiate $S(x)$ with respect to $x$. Using the power rule, if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. So, $S^\prime(x)=2x-\frac{432}{x^{2}}$.

Step4: Find the critical points

Set $S^\prime(x) = 0$ to find the critical points: [ \begin{align*} 2x-\frac{432}{x^{2}}&=0\ 2x&=\frac{432}{x^{2}}\ 2x^{3}&=432\ x^{3}& = 216\ x&=6 \end{align*} ]

Step5: Check the second - derivative

Differentiate $S^\prime(x)$ to get the second - derivative $S^{\prime\prime}(x)=2+\frac{864}{x^{3}}$. When $x = 6$, $S^{\prime\prime}(6)=2+\frac{864}{216}=2 + 4=6>0$. So, the surface area is minimized at $x = 6$.

Step6: Find the height

Substitute $x = 6$ into the formula for $h$: $h=\frac{108}{6^{2}}=\frac{108}{36}=3$.

Answer:

$6,3$