At time seconds a heated metal cube has an edge length of

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

At time

t=0

seconds a heated metal cube has an edge length of

3\text{ in}

. As it warms, the edge length grows so that

\dfrac{dx}{dt}=0.2\sqrt{t}

inches per second, where

t

is measured in seconds and

x

is the edge length in inches. The volume

V

(in cubic inches) of the cube is

V = x^{3}

. (a) Derive an explicit expression for

x

as a function of

t

for

t\ge 0

. State any initial conditions used in your derivation. (b) Using the result from part (a), derive a formula for

\dfrac{dV}{dt}

in terms of

t

. Simplify your expression. (c) Compute

\dfrac{dV}{dt}

t=25\text{ s}

, including units. (d) Determine the time

t

(to the nearest tenth of a second) at which

\dfrac{dV}{dt}=100\text{ in}^3\text{/s}

. Show your analytical work.