Spherical marbles of diameter 1-4 cm are dropped into a 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 ?

Answer

To solve the problem, we need to find out how many spherical marbles of diameter 1.4 cm can be submerged in a beaker with a diameter of 7 cm, which causes the water level to rise by 5.6 cm.

### Step-by-Step Solution:

1. **Determine the radius of the marbles**:
- The diameter of the marbles is given as 1.4 cm.
- Radius of the marbles, \( r = \frac{1.4}{2} = 0.7 \) cm.

2. **Determine the radius of the beaker**:
- The diameter of the beaker is given as 7 cm.
- Radius of the beaker, \( R = \frac{7}{2} = 3.5 \) cm.

3. **Calculate the volume of water displaced**:
- The height by which the water level rises is given as 5.6 cm.
- The volume of water displaced (which is equal to the volume of the marbles submerged) can be calculated using the formula for the volume of a cylinder:
\[ V_{\text{water}} = \pi R^2 h \]
Substituting the values:
\[ V_{\text{water}} = \pi (3.5)^2 (5.6) \]

4. **Calculate the volume of one marble**:
- The volume of a sphere (marble) is given by the formula:
\[ V_{\text{marble}} = \frac{4}{3} \pi r^3 \]
Substituting the radius of the marble:
\[ V_{\text{marble}} = \frac{4}{3} \pi (0.7)^3 \]

5. **Set up the equation to find the number of marbles**:
- Let \( x \) be the number of marbles. The total volume of \( x \) marbles is:
\[ V_{\text{total marbles}} = x \cdot V_{\text{marble}} \]
Since the volume of the displaced water equals the total volume of the marbles, we can write:
\[ \pi R^2 h = x \cdot \frac{4}{3} \pi r^3 \]
Canceling \( \pi \) from both sides:
\[ R^2 h = x \cdot \frac{4}{3} r^3 \]

6. **Solve for \( x \)**:
- Rearranging the equation gives:
\[ x = \frac{R^2 h}{\frac{4}{3} r^3} \]
Substituting the known values:
\[ x = \frac{(3.5)^2 \cdot 5.6}{\frac{4}{3} \cdot (0.7)^3} \]

7. **Calculate the values**:
- Calculate \( R^2 = (3.5)^2 = 12.25 \).
- Calculate \( r^3 = (0.7)^3 = 0.343 \).
- Substitute these into the equation:
\[ x = \frac{12.25 \cdot 5.6}{\frac{4}{3} \cdot 0.343} \]
- Calculate \( \frac{4}{3} \cdot 0.343 = 0.45733 \).
- Now calculate \( x \):
\[ x = \frac{68.4}{0.45733} \approx 149.5 \]
- Since \( x \) must be a whole number, we round it to \( 150 \).

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