Two solid cylinders of radii 4 cm and 5 cm and lengh 6 cm and 4 cm respectively are recast into cylindrical disc of thickness 1 cm. The radius of the disc is

Options

A. a)7 cm
B. b)14 cm
C. c)21 cm
D. d)28 cm

Answer

To solve the problem, we need to find the radius of the cylindrical disc formed by recasting two solid cylinders. We will use the principle that the volume of the two cylinders combined will equal the volume of the disc.

### Step-by-Step Solution:

1. **Calculate the Volume of the First Cylinder:**
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
For the first cylinder with radius \( r_1 = 4 \) cm and height \( h_1 = 6 \) cm:
\[ V_1 = \pi (4^2)(6) = \pi (16)(6) = 96\pi \, \text{cm}^3 \]

2. **Calculate the Volume of the Second Cylinder:**
For the second cylinder with radius \( r_2 = 5 \) cm and height \( h_2 = 4 \) cm:
\[ V_2 = \pi (5^2)(4) = \pi (25)(4) = 100\pi \, \text{cm}^3 \]

3. **Calculate the Total Volume of Both Cylinders:**
The total volume \( V_{total} \) of both cylinders is:
\[ V_{total} = V_1 + V_2 = 96\pi + 100\pi = 196\pi \, \text{cm}^3 \]

4. **Set Up the Volume Equation for the Disc:**
The volume of the cylindrical disc with thickness \( t = 1 \) cm and radius \( r \) is given by:
\[ V_{disc} = \pi r^2 t \]
Substituting \( t = 1 \) cm:
\[ V_{disc} = \pi r^2 (1) = \pi r^2 \, \text{cm}^3 \]

5. **Equate the Volumes:**
Since the volume of the recast material remains constant:
\[ V_{total} = V_{disc} \]
Therefore:
\[ 196\pi = \pi r^2 \]

6. **Solve for \( r^2 \):**
Dividing both sides by \( \pi \):
\[ 196 = r^2 \]

7. **Calculate the Radius \( r \):**
Taking the square root of both sides:
\[ r = \sqrt{196} = 14 \, \text{cm} \]

### Final Answer:
The radius of the disc is \( 14 \) cm.