Question

If radius of circle is increased by 75% then area is increased by ?

Answer 1

Let the radius be x units.

Therefore, Increased radius = \( \Large x + x \times \frac{75}{100} = \frac{7x}{4} \)

Original circumference = \( \Large 2 \pi x \)

Increased circumference = \( \Large 2 \pi \times \frac{7x}{4}=\frac{14 \pi x}{4} \)

= \( \Large \frac{7 \pi x}{2} \)

Therefore, Increase in circumference = \( \Large \frac{7 \pi x}{2}-2 \pi x \)

= \( \Large \frac{3 \pi x}{2} \)

Therefore, Percentage increase = \( \Large \frac{3 \pi \frac{x}{2}}{2 \pi x} \times 100 \)

\( \Large = 75\% \)

Answer 2

$a = \pi {r}^{2} \\ taking \: log \\ log(a) = log(\pi) + 2 log(r) \\ differentiate \\ \frac{da}{a} = 2 \frac{dr}{r} \\ \frac{da}{a} \times 100 = 2 \frac{dr}{r} \times 100 \\ given \: radius \: increased \: by \: 75\% \\ \frac{da}{a} \times 100\%= 2 \times 75\% \\ \frac{da}{a} \times 100\% = 150\% \\ therefore \: area \: increases \: by \: 150\%$

hope it helps.