Question

all edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is 10 centimeters?

Answer 1

Answer:

Volume is changing by $900 \mathrm{cm}^{3} / \mathrm{sec}$

Solution:

As per the problem, all edges of the cube are expanding at a rate of $3 \mathrm{cm} / \mathrm{sec}$

So,$\left(\frac{d s}{d t}\right)$ = $3 \mathrm{cm} / \mathrm{sec}$

We also know that the volume $V=s^{3}$----- (i)

Differentiating the volume from equation (i) we get,

$\frac{d v}{d t}=3 s^{2} \times\left(\frac{d s}{d t}\right)$

$=\left(3 s^{2} \times 3\right)$

$=9 s^{2}$

As given in the problem each edge = 10 cm.

Hence,$\frac{d v}{d t}=9 \times\left(10^{2}\right)$

$=(9 \times 100) \mathrm{cm}^{3} / \mathrm{sec}$

$=900 \mathrm{cm}^{3} / \mathrm{sec}$