Question

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 $\pi = \frac{22}{7}$)

Hint:

Always use fractional forms like $\frac{7}{2}$ instead of decimals like $3.5$ in your calculations.
This makes calculations much easier as many terms will cancel out with $\pi = \frac{22}{7}$.

Answer 1

Step 1: Understanding the Question:
A solid object consists of a central cylindrical part with a hemisphere attached to each of its two flat circular ends.
We are given:
- The total height of the entire solid is $20\text{ cm}$.
- The diameter of the cylinder (and the hemispheres) is $7\text{ cm}$.
We need to find the total volume of this combined solid.

Step 2: Key Formula or Approach:
1. First, find the radius ($r$) of the cylinder and the hemispheres:
\[ r = \frac{\text{Diameter}}{2} = \frac{7}{2} = 3.5\text{ cm} \]
2. Find the height of the cylindrical part ($h$):
The total height includes the height of the cylinder and the radius of each of the two hemispherical ends.
\[ h = \text{Total height} - 2r \]
3. The total volume ($V$) of the solid is the sum of the volumes of the cylinder and the two hemispheres:
\[ V = \text{Volume of Cylinder} + 2 \times \text{Volume of Hemisphere} \]
\[ V = \pi r^2 h + 2 \times \left( \frac{2}{3} \pi r^3 \right) = \pi r^2 h + \frac{4}{3} \pi r^3 = \pi r^2 \left( h + \frac{4}{3}r \right) \]

Step 3: Detailed Explanation:

• 1. Calculate the radius $r$:
\[ r = \frac{7}{2} = 3.5\text{ cm} \]

• 2. Calculate the height of the cylindrical part $h$:
\[ h = 20 - 2(3.5) = 20 - 7 = 13\text{ cm} \]

• 3. Set up the combined volume expression:
\[ V = \pi r^2 \left( h + \frac{4}{3}r \right) \]

• 4. Substitute the values of $r = 3.5 = \frac{7}{2}$, $h = 13$, and $\pi = \frac{22}{7}$:
\[ V = \frac{22}{7} \times \left(\frac{7}{2}\right)^2 \times \left( 13 + \frac{4}{3} \times \frac{7}{2} \right) \]

• 5. Simplify the calculations step-by-step:
- Simplify the radius square term:
\[ \frac{22}{7} \times \frac{49}{4} = \frac{22 \times 7}{4} = \frac{154}{4} = \frac{77}{2} \]
- Simplify the expression inside the brackets:
\[ 13 + \frac{14}{3} = \frac{39 + 14}{3} = \frac{53}{3} \]

• 6. Multiply the simplified terms:
\[ V = \frac{77}{2} \times \frac{53}{3} = \frac{4081}{6}\text{ cm}^3 \]

• 7. Convert the fraction to a decimal or mixed number:
\[ V = 680.17\text{ cm}^3 \text{ (approx) or } 680\frac{1}{6}\text{ cm}^3 \]


Step 4: Final Answer:
The total volume of the solid is $680.17\text{ cm}^3$ (or $\frac{4081}{6}\text{ cm}^3$).