Question

The approximate change in the volume of a cube of side x metre 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

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, $\Delta V=\left(\frac{dV}{dx}\right)\Delta x$
$=\left(3x^2\right)\Delta x=\left(3x^2\right)\left(0.03x\right)$ (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).