Question

A circularmetal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate, if the radius of plate before heating is 10 cm.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{x}\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{circular}\mathrm{plate}\mathrm{and}\mathrm{y}\mathrm{be}\mathrm{its}\mathrm{area}. \\ \mathrm{Now},\mathrm{y}={\mathrm{\pi x}}^2 \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2\mathrm{\pi x} \\ \mathrm{Now},\mathrm{x}=10\mathrm{cm}\left[\mathrm{before}\mathrm{heating}\right] \\ \mathrm{Let}∆\mathrm{x}\mathrm{be}\mathrm{the}\mathrm{increase}\mathrm{in}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{plate}\mathrm{after}\mathrm{heating}\mathrm{and}∆\mathrm{y}\mathrm{be}\mathrm{th}\mathrm{increase}\mathrm{in}\mathrm{its}\mathrm{area}. \\ \mathrm{Now},∆\mathrm{x}=2\%\mathrm{of}10\mathrm{cm}=\frac{2}{100}\times 10=0.2\mathrm{cm} \\ \mathrm{Now},∆\mathrm{y}=\frac{\mathrm{dy}}{\mathrm{dx}}\times ∆\mathrm{x} \\ \Rightarrow ∆\mathrm{y}=2\mathrm{\pi x}\times 0.2=2\mathrm{\pi }\times 10\times 0.2=4\mathrm{\pi }{\mathrm{cm}}^2 \end{gathered}$$