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Differentials, Errors and Approximations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentials, Errors and Approximations5 Marks
Question
A circular metal plate expends under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10cm.
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Answer

Let at any time, x be the radius and y be the area of the plate.
$\text{y}=\text{x}^2$
Let $\triangle\text{x}$ be the change in the radius and $\triangle\text{y}$ be the change in the area of the plate.
We have,
$\frac{\triangle\text{x}}{\text{x}}\times100=\text{k}$
when x = 10, we get
$\triangle\text{x}=\frac{10\text{k}}{100}=\frac{\text{k}}{10}$
Now, $\text{y}=\pi\text{x}^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=2\pi\text{x}$
$\Rightarrow\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}=10\text{cm}}=20\pi\ \text{cm}^2/\text{cm}$
$\therefore\triangle\text{y}=\text{dy}=\frac{\text{dy}}{\text{dx}}\text{dx}=20\pi\times\frac{\text{k}}{10}=2\text{k}\pi\ \text{cm}^2$
Hence, the approximate change in the area of the plate is $2\text{k}\pi\ \text{cm}^2$

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