A spherical ball is melted to form 63 identical cylindrical vessels. If radius of each cylindrical vessel is `33 (1)/(3) %` of radius of spherical ball and height of each cylindrical vessel is 3cm less than radius of each cylindrical vessel, then find radius of spherical ball.

Options

A. A)21cm
B. B)14cm
C. C)35cm
D. D)49cm

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define Variables
Let the radius of the spherical ball be \( R \).

### Step 2: Find the Radius of the Cylindrical Vessel
The radius of each cylindrical vessel is given as \( 33 \frac{1}{3} \% \) of the radius of the spherical ball. This can be expressed as:
\[ \text{Radius of cylindrical vessel} = \frac{33 \frac{1}{3}}{100} \times R = \frac{1}{3} R \]

### Step 3: Find the Height of the Cylindrical Vessel
The height of each cylindrical vessel is 3 cm less than the radius of the cylindrical vessel:
\[ \text{Height of cylindrical vessel} = \frac{1}{3} R - 3 \]

### Step 4: Calculate the Volume of the Spherical Ball
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi R^3 \]

### Step 5: Calculate the Volume of One Cylindrical Vessel
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the radius and height of the cylindrical vessel:
\[ V = \pi \left(\frac{1}{3} R\right)^2 \left(\frac{1}{3} R - 3\right) \]
\[ = \pi \left(\frac{1}{9} R^2\right) \left(\frac{1}{3} R - 3\right) \]
\[ = \frac{\pi}{9} R^2 \left(\frac{1}{3} R - 3\right) \]

### Step 6: Calculate the Total Volume of 63 Cylindrical Vessels
The total volume of 63 cylindrical vessels is:
\[ \text{Total Volume} = 63 \times \frac{\pi}{9} R^2 \left(\frac{1}{3} R - 3\right) \]
\[ = 7 \pi R^2 \left(\frac{1}{3} R - 3\right) \]

### Step 7: Set the Volumes Equal
Since the spherical ball is melted to form the cylindrical vessels, we set the volume of the sphere equal to the total volume of the cylindrical vessels:
\[ \frac{4}{3} \pi R^3 = 7 \pi R^2 \left(\frac{1}{3} R - 3\right) \]

### Step 8: Simplify the Equation
Dividing both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ \frac{4}{3} R^3 = 7 R^2 \left(\frac{1}{3} R - 3\right) \]
\[ \frac{4}{3} R^3 = \frac{7}{3} R^3 - 21 R^2 \]

### Step 9: Rearranging the Equation
Multiply through by 3 to eliminate the fraction:
\[ 4R^3 = 7R^3 - 63R^2 \]
\[ 0 = 3R^3 - 63R^2 \]
Factoring out \( 3R^2 \):
\[ 0 = 3R^2(R - 21) \]

### Step 10: Solve for R
Setting each factor to zero gives:
\[ 3R^2 = 0 \quad \text{or} \quad R - 21 = 0 \]
Since \( R = 0 \) is not a valid solution, we have:
\[ R = 21 \]

### Conclusion
The radius of the spherical ball is \( 21 \) cm.

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