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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES3 Marks
Question
Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are $3\ cm$ and $3.0005\ cm,$ respectively.
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Answer

Let internal radius $= r$ and external radius $= R$
$\therefore$ Volume of hollow spherical shell,
$\text{V}=\frac{4}{3}\pi(\text{R}^{3}-\text{r}^3)$
$\text{V}=\frac{4}{3}\pi\big[(3.0005)^3-(3)^3\big]\ \ \dots(\text{i})$
Now$,$ approximate value of $(3.0005)^3$ can be obtained using differentiation
Let $(3.0005)^3=\text{y}+\triangle\text{y}$
And $\text{x}=3,\triangle\text{x}=0.0005$
Also$,$ let $y = x^3$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=3\text{x}^2$
$\therefore\ \triangle\text{y}=\frac{\text{dy}}{\text{dx}}\times\triangle\text{x}=3\text{x}^3\times0.0005=3\times3^2\times0.0005$
$=27\times0.0005=0.0135$
$\therefore\ (3.0005)^3=\text{y}+\triangle\text{y}=3^3+0.0135=27.0135$
$\therefore\ \text{V}=\frac{4}{3}\pi[27.0135-27000]$
$=\frac{4}{3}\pi[0.0135]=4\pi\times(0.0045)=0.0180\pi\text{cm}^3$

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