24 spheres of the same size are made by melting a solid cylinder, whose diameter is 28 cm and height is 7 cm. The surface area of each sphere is:
(Take `pi=22/7`)

Options

A. `156cm^(2)`
B. `154cm^(2)`
C. `144cm^(2)`
D. `164cm^(2)`

Answer

To find the surface area of each sphere made from melting a solid cylinder, we can follow these steps:

### Step 1: Calculate the volume of the cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Diameter of the cylinder = 28 cm, so the radius \( r = \frac{28}{2} = 14 \) cm.
- Height \( h = 7 \) cm.

Substituting the values into the formula:
\[ V = \pi (14)^2 (7) \]
\[ V = \pi \times 196 \times 7 \]
\[ V = 1372\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of each sphere
Since 24 spheres are made from the cylinder, the volume of each sphere \( V_s \) is:
\[ V_s = \frac{V}{24} = \frac{1372\pi}{24} \]
\[ V_s = \frac{1372}{24} \pi \text{ cm}^3 \]
\[ V_s = \frac{343}{6} \pi \text{ cm}^3 \]

### Step 3: Relate the volume of the sphere to its radius
The volume \( V_s \) of a sphere is given by:
\[ V_s = \frac{4}{3} \pi r^3 \]
Setting the two expressions for volume equal:
\[ \frac{4}{3} \pi r^3 = \frac{343}{6} \pi \]
Dividing both sides by \( \pi \):
\[ \frac{4}{3} r^3 = \frac{343}{6} \]
Multiplying both sides by \( 6 \):
\[ 8 r^3 = 343 \]
Dividing both sides by \( 8 \):
\[ r^3 = \frac{343}{8} \]
Taking the cube root:
\[ r = \sqrt[3]{\frac{343}{8}} = \frac{7}{2} \text{ cm} \]

### Step 4: Calculate the surface area of each sphere
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Substituting \( r = \frac{7}{2} \):
\[ A = 4 \pi \left(\frac{7}{2}\right)^2 \]
\[ A = 4 \pi \left(\frac{49}{4}\right) \]
\[ A = 49\pi \text{ cm}^2 \]
Using \( \pi = \frac{22}{7} \):
\[ A = 49 \times \frac{22}{7} \]
\[ A = 154 \text{ cm}^2 \]

Thus, the surface area of each sphere is **154 cm²**.