# Find the approximate change in the volume V of a cube of side $$x$$ metres caused by increasing the side by $1\%$.

$\begin{array}{1 1} 0.03\; x^3m^3 \\ 0.06\; x^3m^3 \\ 0.05\; x^3m^3 \\ 0.04\; x^3m^3 \end{array}$

Toolbox:
• Let $y=f(x)$
• $\Delta x$ denote a small increment in $x$
• $\Delta y=f(x+\Delta x)-f(x)$
• $dy=\big(\large\frac{dy}{dx}\big)\Delta x$
• Volume of a cube =$l\times b\times h$
Step 1:
Volume =V
Side of the cube=$x$ metres
Increase in side=1%
$\qquad\qquad\;\;\;\;=0.01\times x$
$\qquad\qquad\;\;\;\;=0.01 x$
Volume of a cube =$l\times b\times h$
$\qquad\qquad\qquad=x\times x\times x$
$\qquad\qquad\qquad=x^3$
Step 2:
Approximate change in volume =$\Delta V$
$\qquad\qquad\qquad\qquad\qquad\;\;=\large\frac{dv}{dx}\times \Delta x$
$V=x^3$
$\large\frac{dv}{dx}$$=3x^2[Differentiating with respect to x] \Delta V=\large\frac{dv}{dx}$$\times \Delta x$
$\qquad=3x^2\times 0.01x$
$\qquad=0.03x^3m^3$