Question

On increasing the diameter of a circle by 40%, its area will be increased by
(a) 40%
(b) 80%
(c) 96%
(d) 82%

Answer 1

(c) 96%
Let d be the original diameter.
Radius$=\frac{d}{2}$
Thus, we have:
Original area$=\mathrm{\pi }\times {\left(\frac{d}{2}\right)}^2$
$=\frac{\mathrm{\pi }{d}^2}{4}$
New diameter$=140\%\mathrm{of}d$
$$\begin{gathered} =\left(\frac{140}{100}\times d\right) \\ =\frac{7d}{5} \end{gathered}$$
Now,
New radius$=\frac{7d}{5\times 2}$
$=\frac{7d}{10}$
New area$=\mathrm{\pi }\times {\left(\frac{7d}{10}\right)}^2$
$=\frac{49\mathrm{\pi }{d}^2}{10}$
Increase in the area$=\left(\frac{49\mathrm{\pi }{d}^2}{10}-\frac{\mathrm{\pi }{d}^2}{4}\right)$
$$\begin{gathered} =\frac{24\mathrm{\pi }{d}^2}{100} \\ =\frac{6\mathrm{\pi }{d}^2}{25} \end{gathered}$$
We have:
Increase in the area$=\left(\frac{6\mathrm{\pi }{d}^2}{25}\times \frac{4}{\mathrm{\pi }{d}^2}\times 100\right)\%$
= 96%