A sphere of 30 cm radius is dropped into a cylindrical vessel of 80 cm diameter , which is partly filled with water , then its level rises by x cm . Find x :

Options

A. `27.5 cm `
B. `22.5 cm `
C. `18.5 cm `
D. none of these

Answer

To solve the problem, we need to find the rise in water level (x cm) when a sphere of radius 30 cm is submerged in a cylindrical vessel with a diameter of 80 cm. Here’s a step-by-step solution:

### Step 1: Calculate the Volume of the Sphere
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Given that the radius \( r = 30 \) cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (30)^3 \]

### Step 2: Calculate \( (30)^3 \)
Calculating \( (30)^3 \):
\[ 30^3 = 30 \times 30 \times 30 = 900 \times 30 = 27000 \]
Now substitute this back into the volume formula:
\[ V = \frac{4}{3} \pi (27000) = \frac{108000}{3} \pi = 36000 \pi \text{ cm}^3 \]

### Step 3: Calculate the Radius of the Cylinder
The diameter of the cylindrical vessel is given as 80 cm. Therefore, the radius \( R \) of the cylinder is:
\[ R = \frac{80}{2} = 40 \text{ cm} \]

### Step 4: Set Up the Volume of the Cylinder
The volume of the cylinder can be expressed as:
\[ V = \pi R^2 h \]
where \( R \) is the radius of the cylinder and \( h \) is the height (which is the rise in water level, \( x \) cm). Substituting \( R = 40 \) cm:
\[ V = \pi (40)^2 x = \pi (1600) x = 1600 \pi x \text{ cm}^3 \]

### Step 5: Equate the Volumes
When the sphere is submerged in the cylinder, the volume of water displaced by the sphere is equal to the volume of the cylinder that corresponds to the rise in water level. Therefore, we can set the volumes equal to each other:
\[ 36000 \pi = 1600 \pi x \]

### Step 6: Simplify the Equation
We can divide both sides by \( \pi \) (since \( \pi \) is common and non-zero):
\[ 36000 = 1600 x \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{36000}{1600} \]
Calculating this gives:
\[ x = 22.5 \text{ cm} \]

### Final Answer
Thus, the rise in the water level \( x \) is:
\[ \boxed{22.5 \text{ cm}} \]