The radius of a circle is 9.5 cm. If it increases by 10%, by what percent area will increase?

Question: The radius of a circle is 9.5 cm. If it increases by 10%, by what percent area will increase?

• 21 %
• 16 %
• 10 %
• 20 % 

Answer: 

• Original radius = 9.5 cm
• Increase in radius = 10%

$$\text{Percentage Change in Area} = A + A + \frac{A \times A}{100}$$

📌 $A$ is the percentage change in radius.

 Applying "AB METHOD":

$$= 10 + 10 + \frac{10 \times 10}{100}$$ $$= 10 + 10 + 1$$ $$=21\mathrm{\%}$$

`_` The percentage increase in the area of the circle is 21%.

Alternative: 

Detail Solution :

Given:

• Original radius = 9.5 cm
• Increase in radius = 10%

 Step 1: ✅ Calculate the Original Area-

The area of a circle is given by the formula:

$$A = \pi r^2$$

For the original radius $r = 9.5$ cm,

$$A_{\text{original}} = \pi (9.5)^2$$ $$= \pi \times 90.25$$ $$= 90.25\pi \text{ cm}^2$$

Step 2: ✅ Calculate the New Radius-

Since the radius is increased by 10%,

$$\text{New radius} = 9.5 + \frac{10}{100} \times 9.5$$ $$= 9.5 + 0.95$$ $$= 10.45 \text{ cm}$$

Step 3: ✅ Calculate the New Area-

Using the new radius $r = 10.45$ cm,

$$A_{\text{new}} = \pi (10.45)^2$$ $$= \pi \times 109.2025$$ $$= 109.2025\pi \text{ cm}^2$$

Step 4: ✅ Calculate the Percentage Increase in Area

$$\text{Percentage Increase} = \frac{A_{\text{new}} - A_{\text{original}}}{A_{\text{original}}} \times 100$$ $$= \frac{109.2025\pi - 90.25\pi}{90.25\pi} \times 100$$ $$= \frac{18.9525\pi}{90.25\pi} \times 100$$ $$= \frac{18.9525}{90.25} \times 100$$ $$\approx 21\%$$

'_' The area of the circle increases by 21% when the radius is increased by 10%.

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