A right circular solid cylinder of base radius 4 cm and vertical height 22.5 cm is melted to form 8 equal solid spheres. If there is a process loss of 20% during such formation, then what is the radius of each of the solid sphere so formed?

Options

A. 2 cm
B. 3 cm
C. 2.5 cm
D. 3.5 cm

Answer

To solve the problem, we will 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 of the base of the cylinder.
- \( h \) is the height of the cylinder.

Given:
- Radius \( r = 4 \) cm
- Height \( h = 22.5 \) cm

Substituting the values:

\[ V = \pi (4^2) (22.5) = \pi (16) (22.5) = 360 \pi \text{ cm}^3 \]

### Step 2: Account for the process loss.
There is a process loss of 20%. This means that only 80% of the volume of the cylinder will be used to form the spheres.

Calculating 80% of the volume:

\[ \text{Remaining Volume} = 80\% \text{ of } 360 \pi = 0.8 \times 360 \pi = 288 \pi \text{ cm}^3 \]

### Step 3: Calculate the volume of one solid sphere.
Since the remaining volume is used to form 8 equal solid spheres, the volume of one sphere \( V_s \) is:

\[ V_s = \frac{\text{Total Volume}}{8} = \frac{288 \pi}{8} = 36 \pi \text{ cm}^3 \]

### Step 4: Use the volume of a sphere formula to find the radius.
The formula for the volume \( V \) of a sphere is:

\[ V = \frac{4}{3} \pi r^3 \]

Setting the volume of one sphere equal to \( 36 \pi \):

\[ \frac{4}{3} \pi r^3 = 36 \pi \]

Dividing both sides by \( \pi \):

\[ \frac{4}{3} r^3 = 36 \]

Now, multiply both sides by \( \frac{3}{4} \):

\[ r^3 = 36 \times \frac{3}{4} = 27 \]

### Step 5: Solve for \( r \).
Taking the cube root of both sides:

\[ r = \sqrt[3]{27} = 3 \text{ cm} \]

Thus, the radius of each solid sphere is **3 cm**.

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