Question

The approximate change in the volume of a cube of side $xm$ caused by increasing the side by $3\%$ is

• A. $0.06x^3{m}^3$
• B. $0.6x^3{m}^3$
• C. $0.09x^3{m}^3$
• D. $0.9x^3{m}^3$

Answer 1

Answer: (C)

We know that, the volume $V$ of a cube of side $x$ is given by $V=x^3$
$\Rightarrow \frac{dV}{dx}=3x^2$
Let $\Delta x$ be change in side $=3\%$ of $x=0.03x$
Now, change in volume, $\Delta V=\left(\frac{dV}{dx}\right)\Delta x$
$=\left(3x^2\right)\Delta x=\left(3x^2\right)(0.03x)$ (as $\Delta x=3\%$ of $x$ is $0.03x)$
$=0.09x^3{m}^3$
Hence, the approximate change in the volume of the cube is $0.09x^3{m}^3$.
$\therefore$ The correct option is (c).