35. an open rectangular box with square base is to be made from 48 ft.² of material. what dimensions will…

Explanation:

Step1: Define variables

Let the side - length of the square base be $x$ (in feet) and the height of the box be $h$ (in feet). The surface area $S$ of an open - top rectangular box with a square base is $S = x^{2}+4xh$. Given that $S = 48$, so $x^{2}+4xh=48$, and we can express $h=\frac{48 - x^{2}}{4x}$.

Step2: Express the volume function

The volume $V$ of the box is $V=x^{2}h$. Substitute $h=\frac{48 - x^{2}}{4x}$ into the volume formula: $V(x)=x^{2}\cdot\frac{48 - x^{2}}{4x}=\frac{48x - x^{3}}{4}=12x-\frac{1}{4}x^{3}$, where $x>0$.

Step3: Find the derivative of the volume function

Differentiate $V(x)$ with respect to $x$. Using the power rule, $V^\prime(x)=12-\frac{3}{4}x^{2}$.

Step4: Find the critical points

Set $V^\prime(x) = 0$. Then $12-\frac{3}{4}x^{2}=0$. Rearrange the equation: $\frac{3}{4}x^{2}=12$, $x^{2}=16$, so $x = 4$ (we ignore $x=-4$ since $x$ represents a length and must be positive).

Step5: Find the second - derivative of the volume function

Differentiate $V^\prime(x)$ to get the second - derivative $V^{\prime\prime}(x)=-\frac{3}{2}x$. When $x = 4$, $V^{\prime\prime}(4)=-\frac{3}{2}\times4=-6<0$, which means the volume function $V(x)$ has a maximum at $x = 4$.

Step6: Find the height

Substitute $x = 4$ into the formula for $h$: $h=\frac{48-4^{2}}{4\times4}=\frac{48 - 16}{16}=\frac{32}{16}=2$.

Answer:

The base of the box has side - length $x = 4$ ft and the height $h = 2$ ft.