Question

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is $20\text{ cm}$ and the diameter of the cylinder is $7\text{ cm}$. Find the total volume of the solid. (Use $\pi = \frac{22}{7}$)

Hint:

Always factor out common terms like $\pi r^2$ before plugging in numbers.
This significantly reduces the number of multiplications and divisions you have to perform, keeping your calculations clean and reducing mistakes.

Answer 1

Step 1: Understanding the Question:
The topic is Surface Areas and Volumes of combination of solids.
The solid is made up of three parts: a central cylinder and two hemispherical ends of the same radius.
We are given the total height of the combined solid and the diameter.
We need to calculate the total volume of this compound solid.

Step 2: Key Formula or Approach:
The total volume ($V$) is the sum of the volumes of the individual parts:
\[ V = \text{Volume of Cylinder} + 2 \times \text{Volume of Hemisphere} \]
The relevant formulas are:

• Volume of a Cylinder = $\pi r^2 h$

• Volume of a Hemisphere = $\frac{2}{3}\pi r^3$

We must find the height of the cylindrical part ($h$) by subtracting the heights (radii) of the two hemispherical ends from the total height.

Step 3: Detailed Explanation:

• Identify the given parameters:
Diameter of the cylinder and hemispheres, $D = 7\text{ cm}$
Radius, $r = \frac{D}{2} = \frac{7}{2} = 3.5\text{ cm}$
Total height of the solid = $20\text{ cm}$

• Calculate the height of the cylindrical part ($h$):
Each hemispherical end has a height equal to its radius $r = 3.5\text{ cm}$.
The height of the cylindrical part is the total height minus the radii of the two hemispheres:
\[ h = \text{Total Height} - 2r \]
\[ h = 20 - 2(3.5) = 20 - 7 = 13\text{ cm} \]

• Write down the formula for the total volume of the solid:
\[ V = \text{Volume of Cylinder} + 2 \times \text{Volume of Hemisphere} \]
\[ V = \pi r^2 h + 2 \times \left( \frac{2}{3}\pi r^3 \right) \]
\[ V = \pi r^2 h + \frac{4}{3}\pi r^3 \]
Factor out common terms to simplify:
\[ V = \pi r^2 \left( h + \frac{4}{3}r \right) \]

• Substitute the values ($\pi = \frac{22}{7}$, $r = \frac{7}{2}$, and $h = 13$) into the simplified expression:
\[ V = \frac{22}{7} \times \left( \frac{7}{2} \right)^2 \times \left( 13 + \frac{4}{3} \times \frac{7}{2} \right) \]
\[ V = \frac{22}{7} \times \frac{49}{4} \times \left( 13 + \frac{14}{3} \right) \]

• Simplify the outer factors:
\[ \frac{22}{7} \times \frac{49}{4} = \frac{11 \times 7}{2} = \frac{77}{2} \]

• Simplify the expression inside the bracket:
\[ 13 + \frac{14}{3} = \frac{39 + 14}{3} = \frac{53}{3} \]

• Compute the final volume:
\[ V = \frac{77}{2} \times \frac{53}{3} \]
\[ V = \frac{4081}{6} \approx 680.17\text{ cm}^3 \]


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