Question

question: you have been asked to design a rectangular box with a square base and an open - top. the volume of the box must be 32 cm³. determine the minimum surface area necessary to construct a box of this volume. enter an exact answer. provide your answer below: cm²

Explanation:

Step1: Define variables

Let the side - length of the square base be $x$ cm and the height of the box be $h$ cm. The volume $V=x^{2}h$, and we know $V = 32$, so $h=\frac{32}{x^{2}}$. The surface area $S$ of the open - top box is $S=x^{2}+4xh$.

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

Substitute $h=\frac{32}{x^{2}}$ into $S$: $S=x^{2}+4x\cdot\frac{32}{x^{2}}=x^{2}+\frac{128}{x},x>0$.

Step3: Find the derivative of $S$

Differentiate $S$ with respect to $x$. Using the power rule, if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. So $S^\prime=2x-\frac{128}{x^{2}}$.

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

Set $S^\prime = 0$:
\[

$$\begin{align*} 2x-\frac{128}{x^{2}}&=0\\ 2x&=\frac{128}{x^{2}}\\ 2x^{3}&=128\\ x^{3}&=64\\ x& = 4 \end{align*}$$

\]

Step5: Find the second - derivative of $S$

Differentiate $S^\prime=2x-\frac{128}{x^{2}}$ with respect to $x$ to get the second - derivative $S^{\prime\prime}=2+\frac{256}{x^{3}}$. When $x = 4$, $S^{\prime\prime}=2+\frac{256}{64}=2 + 4=6>0$. So $S$ has a minimum at $x = 4$.

Step6: Calculate the minimum surface area

Substitute $x = 4$ into the surface - area formula $S=x^{2}+\frac{128}{x}$. Then $S=4^{2}+\frac{128}{4}=16 + 32=48$.

Answer:

$48$ cm$^{2}$