A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use `F = 22/7`)

Answer

To find the total volume of a solid that is in the form of a cylinder with hemispherical ends, we will follow these steps:

### Step 1: Identify the dimensions
- The total height of the solid is given as 20 cm.
- The diameter of the cylinder is given as 7 cm.

### Step 2: Calculate the radius
- The radius \( r \) of the cylinder (and the hemispherical ends) can be calculated using the formula:
\[ r = \frac{\text{diameter}}{2} = \frac{7 \text{ cm}}{2} = 3.5 \text{ cm} \]

### Step 3: Calculate the height of the cylinder
- The total height of the solid consists of the height of the cylinder plus the heights of the two hemispherical ends (which together form a full sphere).
- The height of one hemisphere is equal to its radius, which is \( 3.5 \text{ cm} \).
- Therefore, the height of both hemispheres combined is \( 3.5 \text{ cm} + 3.5 \text{ cm} = 7 \text{ cm} \).
- Now, we can find the height of the cylinder \( h \):
\[ h = \text{Total height} - \text{Height of both hemispheres} = 20 \text{ cm} - 7 \text{ cm} = 13 \text{ cm} \]

### Step 4: Calculate the volume of the cylinder
- The volume \( V_c \) of the cylinder can be calculated using the formula:
\[ V_c = \pi r^2 h \]
- Substituting the values:
\[ V_c = \frac{22}{7} \times \left(3.5\right)^2 \times 13 \]
- Calculating \( (3.5)^2 = 12.25 \):
\[ V_c = \frac{22}{7} \times 12.25 \times 13 \]
- Now, calculate \( 12.25 \times 13 = 159.25 \):
\[ V_c = \frac{22}{7} \times 159.25 \]
- Now, calculate \( \frac{22 \times 159.25}{7} = \frac{3503.5}{7} = 500.5 \text{ cm}^3 \)

### Step 5: Calculate the volume of the hemispheres
- The volume \( V_h \) of a single hemisphere is given by:
\[ V_h = \frac{2}{3} \pi r^3 \]
- Therefore, the volume of two hemispheres is:
\[ V_{2h} = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]
- Substituting the radius:
\[ V_{2h} = \frac{4}{3} \times \frac{22}{7} \times \left(3.5\right)^3 \]
- Calculating \( (3.5)^3 = 42.875 \):
\[ V_{2h} = \frac{4}{3} \times \frac{22}{7} \times 42.875 \]
- Now, calculate:
\[ V_{2h} = \frac{4 \times 22 \times 42.875}{3 \times 7} = \frac{3764.5}{21} \approx 179.75 \text{ cm}^3 \]

### Step 6: Calculate the total volume of the solid
- The total volume \( V \) of the solid is the sum of the volume of the cylinder and the volume of the two hemispheres:
\[ V = V_c + V_{2h} = 500.5 \text{ cm}^3 + 179.75 \text{ cm}^3 = 680.25 \text{ cm}^3 \]

### Final Answer
The total volume of the solid is approximately \( 680.25 \text{ cm}^3 \).

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