Question

The approximate change in the volume of a cube of side x metres caused by increasing the side by 2%, is ______________.

Answer 1

Let ∆x be the change in side x and ∆V be the change in the volume of the cube.

It is given that, $\frac{∆x}{x}\times 100=2$ .....(1)

Now,

Volume of the cube of side x, V = x³

$V=x^3$

Differentiating both sides with respect to x, we get

$\frac{dV}{dx}=3x^2$

$\therefore ∆V=\left(\frac{dV}{dx}\right)∆x$

$\Rightarrow ∆V=3x^2∆x$

$\Rightarrow ∆V=3x^2\times \frac{2x}{100}$ [Using (1)]

$\Rightarrow ∆V=\frac{6x^3}{100}$

$\Rightarrow ∆V=0.06x^3$

Thus, the approximate change in volume of the cube is 0.06x³ m³.


The approximate change in the volume of a cube of side x metres caused by increasing the side by 2%, is ___0.06x³ m³___.