Solve.
-An open-top rectangular box has a square base and it will hold 256 cubic centimeters (cc). Each side of the base has length $x \mathrm {~cm}$. The box's surface area $S$ is given by
$$S ( x ) = \frac { 1024 } { x } + x ^ { 2 } .$$
Estimate the minimum surface area and the value of $x$ that will yield it.
A) $256 \mathrm {~cm} ^ { 2 }$ when $x = 6 \mathrm {~cm}$
B) $207 \mathrm {~cm} ^ { 2 }$ when $x = 6 \mathrm {~cm}$
C) $256 \mathrm {~cm} ^ { 2 }$ when $x = 8 \mathrm {~cm}$
D) $192 \mathrm {~cm} ^ { 2 }$ when $\mathrm { x } = 8 \mathrm {~cm}$

Question

Quizplus · Accepted Answer

To find the minimum surface area, we differentiate $S(x)$ with respect to $x$ and set the derivative equal to zero to find critical points. The derivative of $S(x)$ is $S'(x) = -1024/x^2 + 2x$. Setting $S'(x) = 0$ gives $-1024/x^2 + 2x = 0$, which simplifies to $2x^3 = 1024$, leading to $x^3 = 512$, and thus $x = 8$. Substituting $x = 8$ into the original equation for $S(x)$ gives $S(8) = 1024/8 + 8^2 = 128 + 64 = 192 \mathrm{cm}^2$.

Solve -An Open-Top Rectangular Box Has a Square Base and It

Solve
-An Open-Top Rectangular Box Has a Square Base and It