A right triangle contains the right angle between the sides 5 cm and 7 cm. A cone is generated by revolving about the side 5 cm. The volume of this cone is:
Understanding the Cone Volume Problem
The question asks us to find the volume of a cone that is formed by revolving a right triangle around one of its sides that forms the right angle. The lengths of the sides forming the right angle are given as 5 cm and 7 cm. The revolution specifically happens around the side that is 5 cm long.
Identifying Cone Dimensions from Revolution
When a right triangle is revolved around one of its legs (a side forming the right angle), that leg becomes the height of the cone, and the other leg becomes the radius of the circular base of the cone. In this specific problem:
- The side the triangle is revolved about is 5 cm. This side forms the height of the cone. So, the height \(h = 5\) cm.
- The other side forming the right angle is 7 cm. This side forms the radius of the base of the cone. So, the radius \(r = 7\) cm.
The hypotenuse of the right triangle would become the slant height of the cone, but it is not needed to calculate the volume.
Formula for Cone Volume Calculation
The formula used to calculate the volume (\(V\)) of a cone is:
\(V = \frac{1}{3}\pi r^2 h\)
Where:
- \(V\) is the volume of the cone.
- \(\pi\) is the mathematical constant pi (approximately 3.14159).
- \(r\) is the radius of the base of the cone.
- \(h\) is the height of the cone.
Step-by-Step Cone Volume Calculation
Now we substitute the identified values of \(r\) and \(h\) into the volume formula:
- Radius, \(r = 7\) cm.
- Height, \(h = 5\) cm.
- Substitute the values into the formula:
\(V = \frac{1}{3}\pi (7 \text{ cm})^2 (5 \text{ cm})\)
- Calculate the square of the radius: \(7^2 = 49\).
\(V = \frac{1}{3}\pi (49 \text{ cm}^2) (5 \text{ cm})\)
- Multiply the terms: \(49 \times 5 = 245\).
\(V = \frac{1}{3}\pi (245 \text{ cm}^3)\)
- Simplify the expression:
\(V = \frac{245}{3}\pi \text{ cm}^3\)
Converting to a Mixed Number
The result \(\frac{245}{3}\) is an improper fraction. To match the options provided, we convert this to a mixed number.
- Divide 245 by 3.
- \(245 \div 3 = 81\) with a remainder of \(2\).
- So, \(\frac{245}{3}\) can be written as \(81 \frac{2}{3}\).
Therefore, the volume of the cone is \(81 \frac{2}{3}\pi \text{ cm}^3\).
Comparing with Provided Options
Let's compare our calculated volume with the given options:
- Option 1: \(51\frac{2}{3}\pi {\:}c{m^3}\)
- Option 2: \(71\frac{2}{3}\pi {\:}c{m^3}\)
- Option 3: \(81\frac{2}{3}\pi {\:}c{m^3}\)
- Option 4: \(61\frac{2}{3}\pi {\:}c{m^3}\)
Our calculated volume, \(81 \frac{2}{3}\pi \text{ cm}^3\), matches Option 3.
Revision Table: Cone Volume Calculation
| Concept | Description |
|---|---|
| Shape formed | Cone |
| Original Shape | Right triangle (sides 5 cm, 7 cm forming right angle) |
| Axis of Revolution | Side of length 5 cm |
| Cone Height (h) | Length of the side revolved about = 5 cm |
| Cone Radius (r) | Length of the other side forming right angle = 7 cm |
| Volume Formula | \(V = \frac{1}{3}\pi r^2 h\) |
| Calculation | \(V = \frac{1}{3}\pi (7^2)(5) = \frac{1}{3}\pi (49)(5) = \frac{245}{3}\pi\) |
| Result (Mixed Number) | \(81\frac{2}{3}\pi {\:}c{m^3}\) |
Additional Information: Solids of Revolution
When a 2D shape is revolved around an axis, it generates a 3D solid. This is known as a solid of revolution.
- Revolving a rectangle around one of its sides creates a cylinder.
- Revolving a semicircle around its diameter creates a sphere.
- Revolving a right triangle around one of its legs creates a cone.
- Revolving a right triangle around its hypotenuse creates two cones joined at their bases (a double cone).
Understanding which dimension of the 2D shape corresponds to the radius and height of the 3D solid is crucial for calculating volume and surface area.
Important Questions from Solid Figures
The area of the floor of a cubical room is 192 m 2. The length of the longest rod that can be kept in that room is :
A solid metallic rectangular block of dimensions 112 cm × 44 cm × 25 cm is melted and recast into a cylinder of radius 35 cm. The curved surface area (in cm 2) of the cylinder is: (Take π = 22/7)
If the volume of a cube is 175616 cm 3, what is its side?
The volume of a right circular cone is 1232 cm 3. If the height of the cone is 24 cm, then what will be the radius of its base?
The area of the base of a right circular cone is \(\frac{1408}{7} cm^2\) and its height is 6 cm. Taking π = \(\frac{22}{7}\) , the curved surface area of the cone is:
