A cylindrical tub of radius 6 cm, contains water up to a depth of 9 cm. A spherical iron ball is dropped into the tub thus water level reached at 17 cm. Find the radius of the ball.

Options

A. 6 cm
B. 8 cm
C. 4 cm
D. 5 cm

Answer

To find the radius of the spherical iron ball that causes the water level in the cylindrical tub to rise, we can follow these steps:

### Step 1: Understand the Problem
We have a cylindrical tub with a radius of 6 cm and an initial water depth of 9 cm. When a spherical ball is dropped into the tub, the water level rises to 17 cm. We need to find the radius of the ball.

### Step 2: Calculate the Change in Water Level
The initial water level is 9 cm, and after dropping the ball, it rises to 17 cm. Therefore, the increase in water level (h) is:
\[ h = 17 \, \text{cm} - 9 \, \text{cm} = 8 \, \text{cm} \]

### Step 3: Calculate the Volume of Water Displaced
The volume of water displaced by the ball can be calculated using the formula for the volume of a cylinder:
\[ \text{Volume of water displaced} = \pi R^2 h \]
Where:
- \(R\) is the radius of the cylindrical tub (6 cm)
- \(h\) is the increase in water level (8 cm)

Substituting the values:
\[ \text{Volume of water displaced} = \pi (6^2)(8) = \pi (36)(8) = 288\pi \, \text{cm}^3 \]

### Step 4: Calculate the Volume of the Sphere
The volume of the sphere (the iron ball) is given by the formula:
\[ \text{Volume of sphere} = \frac{4}{3} \pi r^3 \]
Where \(r\) is the radius of the sphere.

### Step 5: Set the Volumes Equal
Since the volume of water displaced equals the volume of the sphere, we can set the two volumes equal to each other:
\[ \frac{4}{3} \pi r^3 = 288\pi \]

### Step 6: Simplify and Solve for \(r^3\)
We can cancel \(\pi\) from both sides:
\[ \frac{4}{3} r^3 = 288 \]
Now, multiply both sides by \(\frac{3}{4}\):
\[ r^3 = 288 \times \frac{3}{4} = 216 \]

### Step 7: Calculate the Radius \(r\)
To find \(r\), take the cube root of both sides:
\[ r = \sqrt[3]{216} = 6 \, \text{cm} \]

### Conclusion
The radius of the spherical iron ball is \(6 \, \text{cm}\).

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