a rectangular tank that is 6912 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 $V=x^{2}h$, and we know that $V = 6912$, so $h=\frac{6912}{x^{2}}$.

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 $h=\frac{6912}{x^{2}}$ into the surface - area formula, we get $S=x^{2}+4x\cdot\frac{6912}{x^{2}}=x^{2}+\frac{27648}{x}$.

Step3: Differentiate the surface - area function

Differentiate $S(x)$ with respect to $x$. Using the power rule, $S^\prime(x)=2x-\frac{27648}{x^{2}}$.

Step4: Set the derivative equal to zero

Set $S^\prime(x) = 0$ to find the critical points. So, $2x-\frac{27648}{x^{2}}=0$. Multiply through by $x^{2}$ to get $2x^{3}-27648 = 0$. Then $x^{3}=13824$, and $x=\sqrt[3]{13824}=24$.

Step5: Find the height

Substitute $x = 24$ into the formula for $h$: $h=\frac{6912}{24^{2}}=\frac{6912}{576}=12$.

Answer:

$24,12$