A medicine -capsule is the shape of a cylinder of diameter 0.5 cm with two hemisphere stuck to each of its ends. The length of entire capsule is 2cm. The capacity of the capsule is

Options

A. `0.36 cm^(3)`
B. `0.35 cm^(3)`
C. `0.34 cm^(3)`
D. `0.33 cm^(3)`

Explanation

To find the capacity of the medicine capsule, which is in the shape of a cylinder with two hemispheres at each end, we need to calculate the total volume of the capsule. Here’s how we can do that step by step:

### Step 1: Identify the dimensions
- The diameter of the capsule is given as 0.5 cm.
- Therefore, the radius \( r \) of the capsule is:
\[ r = \frac{\text{diameter}}{2} = \frac{0.5}{2} = 0.25 \text{ cm} \]
- The total length of the capsule is 2 cm.

### Step 2: Determine the height of the cylindrical part
- The height of the cylindrical part can be found by subtracting the heights of the two hemispheres from the total length.
- The height of each hemisphere is equal to the radius, which is 0.25 cm.
- Therefore, the total height of the two hemispheres is:
\[ \text{Height of two hemispheres} = 2 \times 0.25 = 0.5 \text{ cm} \]
- Now, we can find the height of the cylindrical part \( h \):
\[ h = \text{Total length} - \text{Height of two hemispheres} = 2 - 0.5 = 1.5 \text{ cm} \]

### Step 3: Calculate the volume of the cylindrical part
- The volume \( V_c \) of the cylinder is given by the formula:
\[ V_c = \pi r^2 h \]
- Substituting the values:
\[ V_c = \pi (0.25)^2 (1.5) = \pi (0.0625) (1.5) = \pi (0.09375) \]
- Using \( \pi \approx \frac{22}{7} \):
\[ V_c \approx \frac{22}{7} \times 0.09375 \approx 0.2207 \text{ cm}^3 \]

### Step 4: Calculate the volume of the two hemispheres
- The volume \( V_h \) of one hemisphere is given by the formula:
\[ V_h = \frac{2}{3} \pi r^3 \]
- Therefore, the volume of two hemispheres is:
\[ V_h = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]
- Substituting the radius:
\[ V_h = \frac{4}{3} \pi (0.25)^3 = \frac{4}{3} \pi (0.015625) = \frac{4 \times 0.015625}{3} \pi \]
- Calculating:
\[ V_h \approx \frac{0.0625}{3} \pi \approx 0.0208 \text{ cm}^3 \]

### Step 5: Calculate the total volume of the capsule
- The total volume \( V \) of the capsule is the sum of the volume of the cylinder and the volume of the two hemispheres:
\[ V = V_c + V_h \]
- Substituting the values:
\[ V \approx 0.2207 + 0.0208 \approx 0.2415 \text{ cm}^3 \]

### Conclusion
The capacity of the capsule is approximately:
\[ \text{Capacity} \approx 0.2415 \text{ cm}^3 \]