All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume changing when each edge is 5 centimeters?

Question

Quizplus · Accepted Answer

The volume of a cube is given by $V = s^3$, where $s$ is the length of an edge. The rate of change of volume with respect to time ($\frac{dV}{dt}$) can be found by differentiating both sides with respect to $t$, giving $\frac{dV}{dt} = 3s^2 \cdot \frac{ds}{dt}$. Given $\frac{ds}{dt} = 6$ cm/sec and $s = 5$ cm, substituting these values gives $\frac{dV}{dt} = 3 \cdot 5^2 \cdot 6 = 450$ cm³/sec.

All Edges of a Cube Are Expanding at a Rate