Question

# The volume of a cube is increasing at the rate

Solid Geometry
The volume of a cube is increasing at the rate of 1200 cm³/min at the instant its edges are 20 cm long. At what rate are the edges changing at that instant?

2021-08-10

The volume of a cube with edge length ee is given by:
$$\displaystyle{V}={e}^{{3}}$$
Differentiate with respect to t: $$\frac{dV}{dt}=3e^2\frac{de}{dt}$$
Substitute $$\frac{dV}{dt}=1200\ cm^3/min$$ and e=20 cm then solve for de/dt: $$\displaystyle{1200}={3}{\left({20}\right)}^{{2}}\cdot{\left({}\frac{{de}}{{\left.{d}{t}\right.}}\right)}$$
$$\displaystyle{1200}={1200}{\left({}\frac{{de}}{{\left.{d}{t}\right.}}\right)}$$
$$\displaystyle{1}={}\frac{{de}}{{\left.{d}{t}\right.}}$$
or
$$\displaystyle{}\frac{{de}}{{\left.{d}{t}\right.}}={1}{}\frac{{cm}}{\min}$$