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Provide solution for RD Sharma maths class 12 chapter 13 Differentials Errors and Approximations exercise  multiple choice question 8

Answers (1)

Answer: \text { (c) } 3 \lambda \%

Hint: Here, we all know the formula about volume of cone

 

                V=\frac{1}{3} \pi r^{2} h

 

Given:

                \frac{r}{h}=\frac{1}{2}

So, volume of cone  V=\frac{1}{3} \pi r^{2} h

Solution:

So,

            \begin{aligned} &V=\frac{1}{3} \pi r^{2}(2 r) \\\\ &V=\frac{2}{3} \pi r^{3} \end{aligned}

Let’s differentiate,

            \begin{aligned} &V=\frac{2}{3} \pi r^{3} \\\\ &\frac{d V}{d r}=2 \pi r^{2} \end{aligned}

Percentage error in measuring=\lambda \%

            \begin{aligned} &\frac{\Delta r}{r}=\frac{\lambda}{100} \\\\ &\Delta r=\lambda r(100) \end{aligned}

Approximate error in V=dV

            \begin{aligned} &=\frac{d V}{d r} \times \Delta r \\\\ &=\frac{\lambda}{100}\left(2 \pi r^{3}\right) \end{aligned}

            \begin{aligned} &=\frac{3 \lambda}{100}\left(\frac{2}{3} \pi r^{3}\right) \\\\ &=3 \lambda \% \text { of } V \end{aligned}

So, percentage error in V=3\lambda \%

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