The surface area of a solid sphere is increased by 21% without changing its shape. Find the percentage increase in its:
radius

Answer

To find the percentage increase in the radius of a solid sphere when its surface area is increased by 21%, we can follow these steps:

### Step 1: Understand the formula for the 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: Calculate the original surface area
Let the original radius of the sphere be \( r \). Therefore, the original surface area is:
\[ A = 4\pi r^2 \]

### Step 3: Calculate the new surface area after the increase
The surface area is increased by 21%. Thus, the new surface area \( A' \) can be calculated as:
\[ A' = A + 0.21A = 4\pi r^2 + 0.21 \times 4\pi r^2 = 4\pi r^2 (1 + 0.21) = 4\pi r^2 \times 1.21 \]
This simplifies to:
\[ A' = 4.84\pi r^2 \]

### Step 4: Relate the new surface area to the new radius
Let the new radius be \( R \). The surface area of the sphere with the new radius is:
\[ A' = 4\pi R^2 \]
Setting the two expressions for the new surface area equal gives:
\[ 4\pi R^2 = 4.84\pi r^2 \]
Dividing both sides by \( 4\pi \) yields:
\[ R^2 = 4.84 r^2 \]

### Step 5: Solve for the new radius
Taking the square root of both sides:
\[ R = \sqrt{4.84} r \]
Calculating the square root:
\[ R = 2.2 r \]

### Step 6: Calculate the percentage increase in radius
The percentage increase in radius is given by:
\[ \text{Percentage Increase} = \left( \frac{R - r}{r} \right) \times 100 \]
Substituting \( R = 2.2r \):
\[ \text{Percentage Increase} = \left( \frac{2.2r - r}{r} \right) \times 100 = \left( \frac{1.2r}{r} \right) \times 100 = 120\% \]

### Final Answer
The percentage increase in the radius of the sphere is **120%**.

---