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.

\[\text { Then,} \]

\[ y = x^2 \]

\[\text { Let ∆ x be the change in the radius and }\bigtriangleup y \text { be the change in the area of the plate }. \]

\[\text { We have }\]

\[\frac{∆ x}{x} \times 100 = k\]

\[\text { When }x = 10,\text {  we get }\]

\[ ∆ x = \frac{10k}{100} = \frac{k}{10}\]

\[\text { Now,} y = \pi x^2 \]

\[ \Rightarrow \frac{dy}{dx} = 2\pi x\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 10 cm} = 20\pi {cm}^2 /cm\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = 20\pi \times \frac{k}{10} = 2k\pi \ {cm}^2 \]

Hence, the approximate change in the area of the plate is 2k 

\[\pi\] cm² .