$\text { The volume of a cube with sides of length } s \text { is given by } V = s ^ { 3 } \text {. Find the rate of }$ change of volume with respect to $s$ when $s = 6$ centimeters.
A) $648 \mathrm {~cm} ^ { 2 }$
B) $216 \mathrm {~cm} ^ { 2 }$
C) $36 \mathrm {~cm} ^ { 2 }$
D) $108 \mathrm {~cm} ^ { 2 }$
E) $72 \mathrm {~cm} ^ { 2 }$

Question

Quizplus · Accepted Answer

We need to find the rate of change of volume with respect to $s$, which is the derivative of the function $V(s) = s^3$ with respect to $s$.
\begin{align*}
\frac{dV}{ds} &= 3s^2\
\end{align*}
When $s = 6$, we have
\begin{align*}
\frac{dV}{ds}\Big|_{s=6} &= 3(6)^2\
&= 108\
\end{align*}
Therefore, the rate of change of volume with respect to $s$ when $s = 6$ centimeters is $108 \mathrm{~cm}^2$. Hence, the answer is D.

The volume of a cube with sides of length s is given by V=s3. Find the rate of \text { The volume of a cube with sides of length } s \text { is given by } V = s ^ { 3 } \text {. Find the rate of } The volume of a cube with sides of length s is given by V=s3. Find the rate of

$\text { The volume of a cube with sides of length } s \text { is given by } V = s ^ { 3 } \text {. Find the rate of }$