If the radius of a sphere is increased by 3%, then what per cent increase takes place in surface area of the sphere?

Options

A. 0.0609
B. 0.07
C. 0.0506
D. 0.09

Answer

To find the percentage increase in the surface area of a sphere when the radius is increased by 3%, we can follow these steps:

### Step 1: Understand the formula for surface area of a sphere
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Determine the new radius after the increase
If the radius is increased by 3%, the new radius \( r' \) can be calculated as:
\[ r' = r + 0.03r = 1.03r \]

### Step 3: Calculate the new surface area
Using the new radius, the new surface area \( A' \) can be calculated as:
\[ A' = 4\pi (r')^2 = 4\pi (1.03r)^2 = 4\pi (1.0609r^2) = 4\pi \cdot 1.0609 \cdot r^2 \]

### Step 4: Express the new surface area in terms of the original surface area
The original surface area \( A \) is:
\[ A = 4\pi r^2 \]
Thus, we can express the new surface area as:
\[ A' = 1.0609 \cdot A \]

### Step 5: Calculate the increase in surface area
The increase in surface area is given by:
\[ \text{Increase} = A' - A = 1.0609A - A = 0.0609A \]

### Step 6: Calculate the percentage increase in surface area
To find the percentage increase, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{A} \right) \times 100\% \]
Substituting the increase we found:
\[ \text{Percentage Increase} = \left( \frac{0.0609A}{A} \right) \times 100\% = 6.09\% \]

### Final Answer
The percentage increase in the surface area of the sphere is **6.09%**.

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