Question

Find the approximate change in the volume $V$ of a cube of side $x$ meters caused by increasing the side by $2\%$.

• A. $1.06x^3\,m^3$
• B. $1.26x^3\,m^3$
• C. $2.50x^3\,m^3$
• D. $0.06x^3\,m^3$

Answer 1

Answer: (D)

$0.06x^3\,m^3$

Answer 2

It is given that $\frac{\Delta x}{x} \times 100 =2$ We have, $V = x^{3}$ $\Rightarrow \frac{dV}{dx} = 3x^{2}$ $\therefore \Delta V = \frac{dV}{dx} \times\Delta x$ $\Rightarrow \Delta V = 3x^{2}\,\Delta x$ $\Rightarrow \Delta V = 3x^{2} \times \frac{2x}{100}$ $\Rightarrow \Delta V = 0.06x^{2}$ Thus, the approximate change in volume is $0.06x^3\, m^3$.