#### Please solve RD Sharma class 12 chapter Differentials Errors and Approximations exercise 13.1 question 5 maths textbook solution

Hint: Surface volume of a sphere of radius $x$ is given by $v=\frac{4}{3} \pi x^{3}$

Given: let  $x$ be the radius and $\Delta x$ be the error in the value $x$

Solution: Suppose $x$ be the radius of the sphere and $\Delta x$ be the error in the value $x$

⇒ Thus, we have $\Delta x=\left(\frac{0.1}{100}\right) \times(x)$

So,   $\Delta x=0.001 x$

⇒ Volume of a sphere,  $v=\frac{4}{3} \pi x^{3}$

So, Differentiate v with respect $x$

\begin{aligned} &\Rightarrow \frac{d v}{d x}=\frac{d}{d x}\left(\frac{4}{3} \pi x^{3}\right) \Rightarrow \frac{d v}{d x}=\frac{4 \pi}{3} \frac{d}{d x}\left(x^{3}\right) \\\\ \\\\ &\Rightarrow \frac{d v}{d x}=\frac{4 \pi}{3}\left(4 x^{2}\right)=4 \pi x^{2} \end{aligned}

$\Rightarrow$ As we know, $y=f(x)$  and $\Delta x$  is a smaller increment,

\begin{aligned} &\Delta y=\left(\frac{d v}{d x}\right) \Delta x=0.001 x \\\\ &\Delta x=0.004 \pi x^{3} \\\\ &\text { Here, } \frac{d v}{d x}=4 \pi x^{2} \text { and } \Delta x=0.001 x \end{aligned}

$\Rightarrow$ Percentage error    $=\frac{0.004 \pi x^{3}}{\frac{4}{3} \pi x^{3}} \times 100=0.003 \times 100=0.3 \%$