Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-The function $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 3 }$ describes the volume of a cube, $\mathrm { f } ( \mathrm { x } )$, in cubic inches, whose length, width, and height each measure $x$ inches. Find the instantaneous rate of change of the volume with respect to $x$ when $x = 4$ inches.
A) 12 cubic inches per inch
B) $49.21$ cubic inches per inch
C) 48 cubic inches per inch
D) 192 cubic inches per inch

Question

Quizplus · Accepted Answer

The instantaneous rate of change of the volume with respect to $x$ is found by taking the derivative of the volume function $V(x) = x^3$ with respect to $x$, which gives $V'(x) = 3x^2$. Substituting $x = 4$ inches into $V'(x)$ gives $V'(4) = 3(4)^2 = 3(16) = 48$ cubic inches per inch.

Choose the One Alternative That Best Completes the Statement or Answers