Marbles of diameter 1.4 cm are dropped into a cylindrical beaker containing some water and are fully submerged. The diameter of the beaker is 7 cm. Find how many marbles have been dropped in it if the water rises by 5.6 cm.

Options

A. 50
B. 150
C. 250
D. 350

Answer

To solve the problem, we need to find out how many marbles are dropped into the beaker based on the rise in water level. Here are the steps to find the solution:

### Step 1: Calculate the volume of water displaced by the rise in water level.

The volume of water displaced can be calculated using the formula for the volume of a cylinder:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the beaker
- \( h \) is the height the water rises

Given:
- Diameter of the beaker = 7 cm, so the radius \( r = \frac{7}{2} = 3.5 \) cm
- Height the water rises \( h = 5.6 \) cm

Substituting the values:

\[ V = \pi (3.5)^2 (5.6) \]

Calculating \( (3.5)^2 = 12.25 \):

\[ V = \pi \times 12.25 \times 5.6 \]

Calculating \( 12.25 \times 5.6 = 68.8 \):

\[ V = \pi \times 68.8 \]

Using \( \pi \approx 3.14 \):

\[ V \approx 3.14 \times 68.8 \approx 216.592 \text{ cm}^3 \]

### Step 2: Calculate the volume of one marble.

The volume of a sphere is given by the formula:

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

Where \( r \) is the radius of the marble. Given the diameter of the marble is 1.4 cm, the radius \( r = \frac{1.4}{2} = 0.7 \) cm.

Substituting the value:

\[ V = \frac{4}{3} \pi (0.7)^3 \]

Calculating \( (0.7)^3 = 0.343 \):

\[ V = \frac{4}{3} \pi \times 0.343 \]

Calculating \( \frac{4}{3} \times 0.343 \approx 0.4573 \):

\[ V \approx 0.4573 \pi \]

Using \( \pi \approx 3.14 \):

\[ V \approx 0.4573 \times 3.14 \approx 1.436 \text{ cm}^3 \]

### Step 3: Calculate the number of marbles.

To find the number of marbles, we divide the total volume of water displaced by the volume of one marble:

\[ \text{Number of marbles} = \frac{\text{Volume of displaced water}}{\text{Volume of one marble}} = \frac{216.592}{1.436} \]

Calculating:

\[ \text{Number of marbles} \approx 150 \]

### Final Answer:
The number of marbles that have been dropped into the beaker is **150**.

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