If the radius of a sphere is increased by 2 cm. Its surface area increased by `352 cm^(2)`. The radius of sphere before change is :
(use `pi = (22)/(7)`)

Options

A. 3 cm
B. 4 cm
C. 5 cm
D. 6 cm

Answer

To solve the problem, we need to find the original radius of the sphere before it was increased by 2 cm. We know that the surface area of a sphere is given by the formula:

\[ \text{Surface Area} = 4\pi r^2 \]

where \( r \) is the radius of the sphere.

### Step 1: Write the equation for the surface area before and after the increase in radius.

Let the original radius of the sphere be \( r \) cm. After increasing the radius by 2 cm, the new radius becomes \( r + 2 \) cm.

The surface area before the increase is:
\[ \text{Surface Area}_1 = 4\pi r^2 \]

The surface area after the increase is:
\[ \text{Surface Area}_2 = 4\pi (r + 2)^2 \]

### Step 2: Set up the equation based on the increase in surface area.

According to the problem, the increase in surface area is 352 cm². Therefore, we can write the equation:
\[ \text{Surface Area}_2 - \text{Surface Area}_1 = 352 \]

Substituting the expressions for the surface areas:
\[ 4\pi (r + 2)^2 - 4\pi r^2 = 352 \]

### Step 3: Factor out the common term.

We can factor out \( 4\pi \) from the left side:
\[ 4\pi \left((r + 2)^2 - r^2\right) = 352 \]

### Step 4: Simplify the expression inside the parentheses.

Now, we need to simplify \( (r + 2)^2 - r^2 \):
\[ (r + 2)^2 = r^2 + 4r + 4 \]
Thus,
\[ (r + 2)^2 - r^2 = (r^2 + 4r + 4) - r^2 = 4r + 4 \]

### Step 5: Substitute back into the equation.

Now, substituting back, we have:
\[ 4\pi (4r + 4) = 352 \]

### Step 6: Divide both sides by \( 4\pi \).

To isolate \( r \), we divide both sides by \( 4\pi \):
\[ 4r + 4 = \frac{352}{4\pi} \]

Using \( \pi = \frac{22}{7} \):
\[ 4r + 4 = \frac{352}{4 \times \frac{22}{7}} = \frac{352 \times 7}{88} = \frac{2464}{88} = 28 \]

### Step 7: Solve for \( r \).

Now, we can solve for \( r \):
\[ 4r + 4 = 28 \]
Subtracting 4 from both sides:
\[ 4r = 24 \]
Dividing by 4:
\[ r = 6 \]

### Conclusion

The original radius of the sphere before the change is \( r = 6 \) cm.