Question

Radius of circle is 1m if diameter is increased by 100% then find the percentage increase in its area.

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Answer 1

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Let the diameter be 100 cm.

Let the diameter be 100 cm.New diameter = 200 cm

Let the diameter be 100 cm.New diameter = 200 cmChange in area = π(100)^2−π(50)^2

2=π(100+50)(100−50)

2=π(100+50)(100−50)=π×150×50

2=π(100+50)(100−50)=π×150×50= 7500π cm^2.

2.Percentage increase = 7500ππ×50×50×100 = 300%

Let the diameter be 100 cm.

New diameter = 200 cm

Change in area = π(100)^2−π(50)^2

=π(100+50)(100−50)

=π×150×50

= 7500πcm^2.

Percentage increase = 7500ππ×50×50×100 = 300%

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Answer 2

Given,

Radius of circle, $\[r = 1m\]$

Increase in diameter $\[=100\% \]$

Solution,

Consider the diameter is d.

$\[\begin{array}{l}d = 2 \times r\\ \Rightarrow d = 2 \times 1\\ \Rightarrow d = 2m\end{array}\]$

Initial diameter, $\[{d_1} = 2m\]$

Final diameter,

$\[\begin{array}{l}{d_2} = {d_1} + 100\% {d_1}\\ \Rightarrow {d_2} = 2 + \frac{{100}}{{100}} \times 2\\ \Rightarrow {d_2} = 4m\end{array}\]$

Consider the area is a.

$\[a = \frac{\pi }{4}{d^2}\]$

$\[a \propto {d^2}\]$

$\[\frac{{\Delta a}}{a} \times 100 = 2 \times \frac{{\Delta d}}{d} \times 100\]$

$\[\frac{{\Delta a}}{a} \times 100 = 2 \times \frac{{{d_2} - {d_1}}}{{{d_1}}} \times 100\]$

$\[\frac{{\Delta a}}{a} \times 100 = 2 \times \frac{{4 - 2}}{2} \times 100\]$

$\[\frac{{\Delta a}}{a} \times 100 = 2 \times 1 \times 100\]$

$\[\frac{{\Delta a}}{a} \times 100 = 200\% \]$

Hence percentage increase in area is $\[200\% \]$.