question 1 of 12, step 1 of 1 a rectangular box is designed to have a square base and an open - top. the…

Explanation:

Step1: Define variables

Let the side - length of the square base be $x$ (in inches) and the height of the box be $h$ (in inches). The volume $V$ of the box is $V=x^{2}h$, and we know that $V = 2916$, so $h=\frac{2916}{x^{2}}$. The surface area $S$ of the open - top box is $S=x^{2}+4xh$.

Step2: Substitute $h$ into the surface - area formula

Substitute $h=\frac{2916}{x^{2}}$ into $S$: $S=x^{2}+4x\cdot\frac{2916}{x^{2}}=x^{2}+\frac{11664}{x},x>0$.

Step3: Find the derivative of $S$

Differentiate $S(x)$ with respect to $x$. Using the power rule, if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. So $S^\prime(x)=2x-\frac{11664}{x^{2}}$.

Step4: Set the derivative equal to zero and solve for $x$

Set $S^\prime(x)=0$: [ \begin{align*} 2x-\frac{11664}{x^{2}}&=0\ 2x&=\frac{11664}{x^{2}}\ 2x^{3}&=11664\ x^{3}& = 5832\ x&=18 \end{align*} ]

Step5: Find the second - derivative of $S$

Differentiate $S^\prime(x)$ to get the second - derivative $S^{\prime\prime}(x)=2+\frac{23328}{x^{3}}$. When $x = 18$, $S^{\prime\prime}(18)=2+\frac{23328}{18^{3}}=2 + 4=6>0$. So $x = 18$ gives a minimum of the surface - area function.

Step6: Calculate the minimum surface area

Substitute $x = 18$ into the surface - area formula $S=x^{2}+\frac{11664}{x}$. Then $S=18^{2}+\frac{11664}{18}=324 + 648=972$.

Answer:

972