How many hemispherical balls can be made from a cylinder 56 cm high and 12 cm diameter, when every ball being 0.75 cm in radius?

Options

A. 1792
B. 3584
C. 4824
D. 7168

Answer

To find out how many hemispherical balls can be made from a cylinder, we need to calculate the volume of the cylinder and the volume of one hemispherical ball, and then divide the volume of the cylinder by the volume of one ball.

### Step 1: Calculate the volume of the cylinder
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder

Given:
- Diameter of the cylinder = 12 cm, so the radius \( r = \frac{12}{2} = 6 \) cm
- Height \( h = 56 \) cm

Now, substituting the values into the formula:
\[ V_{\text{cylinder}} = \pi (6)^2 (56) \]
\[ V_{\text{cylinder}} = \pi (36)(56) \]
\[ V_{\text{cylinder}} = 2016\pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of one hemispherical ball
The formula for the volume of a hemisphere is given by:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
Where:
- \( r \) is the radius of the hemisphere

Given:
- Radius of the ball = 0.75 cm

Now, substituting the value into the formula:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (0.75)^3 \]
Calculating \( (0.75)^3 \):
\[ (0.75)^3 = 0.421875 \]
Now substituting this back:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (0.421875) \]
\[ V_{\text{hemisphere}} = \frac{0.84375}{3} \pi \]
\[ V_{\text{hemisphere}} = 0.28125\pi \, \text{cm}^3 \]

### Step 3: Calculate the number of hemispherical balls
To find the number of hemispherical balls that can be made from the cylinder, we divide the volume of the cylinder by the volume of one hemisphere:
\[ \text{Number of balls} = \frac{V_{\text{cylinder}}}{V_{\text{hemisphere}}} \]
Substituting the volumes we calculated:
\[ \text{Number of balls} = \frac{2016\pi}{0.28125\pi} \]
The \( \pi \) cancels out:
\[ \text{Number of balls} = \frac{2016}{0.28125} \]
Calculating this gives:
\[ \text{Number of balls} \approx 7176 \]

### Final Answer:
Approximately 7176 hemispherical balls can be made from the cylinder.

---