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 10 cm.

Answer 1

Let at any time, x be the radius and y be the area of the plate.
$$\begin{gathered} \mathrm{Then}, \\ y=x^2 \\ \mathrm{Let}∆x\mathrm{be}\mathrm{the}\mathrm{change}\mathrm{in}\mathrm{the}\mathrm{radius}\mathrm{and}△y\mathrm{be}\mathrm{the}\mathrm{change}\mathrm{in}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{plate}. \\ \mathrm{We}\mathrm{have} \\ \frac{∆x}{x}\times 100=k \\ \mathrm{When}x=10,weget \\ ∆x=\frac{10k}{100}=\frac{k}{10} \\ \mathrm{Now},y=\mathrm{\pi }{x}^2 \\ \Rightarrow \frac{dy}{dx}=2\mathrm{\pi }x \\ \Rightarrow {\left(\frac{dy}{dx}\right)}_{x=10\mathrm{cm}}=20\mathrm{\pi }{\mathrm{cm}}^2/\mathrm{cm} \\ \therefore ∆y=dy=\frac{dy}{dx}dx=20\mathrm{\pi }\times \frac{k}{10}=2\mathrm{k}\pi {\mathrm{cm}}^2 \end{gathered}$$

Hence, the approximate change in the area of the plate is 2k$\mathrm{\pi }$ cm² .