The radii of the ends of a frustum of a cone 7 cm high are 5 cm and 3 cm. Find its volume correct to one decimal place. (use \(\pi = \frac{{22}}{7}\))
359.3 cm3
Understanding the Frustum of a Cone
A frustum of a cone is the part of a cone that remains when the top part is cut off by a plane parallel to the base. It has two circular bases of different radii and a height.
To find the volume of a frustum of a cone, we use a specific formula that takes into account the radii of both bases and the height of the frustum.
Calculating the Volume of the Frustum
The question asks us to calculate the volume of a frustum of a cone with given dimensions. We are provided with:
- Height (\(h\)) = 7 cm
- Radius of the larger base (\(R\)) = 5 cm
- Radius of the smaller base (\(r\)) = 3 cm
- Value of \(\pi = \frac{{22}}{7}\)
The formula for the volume (\(V\)) of a frustum of a cone is:
\(V = \frac{1}{3}\pi h (R^2 + Rr + r^2)\)
Applying the Formula to Find the Frustum Volume
Now, let's substitute the given values into the formula:
\(V = \frac{1}{3} \times \frac{22}{7} \times 7 \times (5^2 + 5 \times 3 + 3^2)\)
First, calculate the terms inside the parenthesis:
\(R^2 = 5^2 = 25\)
\(Rr = 5 \times 3 = 15\)
\(r^2 = 3^2 = 9\)
Sum these values:
\(R^2 + Rr + r^2 = 25 + 15 + 9 = 49\)
Now, substitute this sum back into the volume formula:
\(V = \frac{1}{3} \times \frac{22}{7} \times 7 \times 49\)
We can cancel out the 7 in the numerator and the denominator:
\(V = \frac{1}{3} \times 22 \times 49\)
Multiply 22 by 49:
\(22 \times 49 = 1078\)
Now, divide by 3:
\(V = \frac{1078}{3}\)
Performing the division:
\(1078 \div 3 \approx 359.333...\)
The question asks for the volume correct to one decimal place. Rounding 359.333... to one decimal place gives 359.3 cm\(^3\).
Summary of Calculations
| Parameter | Value |
|---|---|
| Height (\(h\)) | 7 cm |
| Larger Radius (\(R\)) | 5 cm |
| Smaller Radius (\(r\)) | 3 cm |
| \(\pi\) | \(\frac{{22}}{7}\) |
| Formula | \(V = \frac{1}{3}\pi h (R^2 + Rr + r^2)\) |
| Calculation | \(V = \frac{1}{3} \times \frac{22}{7} \times 7 \times (5^2 + 5 \times 3 + 3^2)\) \(V = \frac{22}{3} \times (25 + 15 + 9)\) \(V = \frac{22}{3} \times 49\) \(V = \frac{1078}{3}\) \(V \approx 359.333...\) |
| Rounded Volume (1 d.p.) | 359.3 cm\(^3\) |
Therefore, the volume of the frustum of the cone is approximately 359.3 cm\(^3\).
Revision Table: Frustum of Cone Formulas
| Aspect | Formula | Description |
|---|---|---|
| Volume | \(V = \frac{1}{3}\pi h (R^2 + Rr + r^2)\) | \(h\) = height, \(R\) = radius of larger base, \(r\) = radius of smaller base |
| Slant Height | \(l = \sqrt{h^2 + (R-r)^2}\) | \(h\) = height, \(R\) = radius of larger base, \(r\) = radius of smaller base |
| Curved Surface Area | \(CSA = \pi l (R+r)\) | \(l\) = slant height, \(R\) = radius of larger base, \(r\) = radius of smaller base |
| Total Surface Area | \(TSA = \pi (R^2 + r^2 + l(R+r))\) | \(R\) = radius of larger base, \(r\) = radius of smaller base, \(l\) = slant height |
Additional Information: Understanding Solid Shapes and Volume
Solid shapes, also known as 3D shapes, occupy space and have volume. Common solid shapes include cubes, cuboids, cylinders, cones, spheres, and frustums.
- Volume: Volume is the amount of three-dimensional space a solid object occupies. It is measured in cubic units (like cm\(^3\), m\(^3\)).
- Cone: A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. Its volume is \(\frac{1}{3}\pi r^2 h\).
- Frustum: As discussed, a frustum is derived from a cone by cutting off the top part parallel to the base. The volume of a frustum can also be thought of as the volume of the original larger cone minus the volume of the smaller cone that was removed. The formula used here is a direct way to calculate it without needing the height of the original cone.
Understanding the formulas for these basic shapes is crucial in geometry and mensuration problems.
Similar Questions
The curved surface area and the volume of a cylindrical object are 88 cm 2and 132 cm 3, respectively. The height (in cm) of the cylindrical object is:
(Take π = \(\frac{{22}}{7}\) )
The circumference of the base of a cylindrical vessel is 264 cm and its height is 50 cm. The capacity (in litres) of the vessel is:
(Take π = \(\frac{22}{7}\) )
What will be the total cost (in Rs.) of polishing the curved surface of a wooden cylinder at rate of Rs. 50 per m 2, if its diameter is 70 cm and height is 6 m? (Take π = \(\frac{22}{7}\) )
A cylindrical tube, open at both ends, is made of a metal sheet which is 0.5 cm thick. Its outer radius is 4 cm and length is 2 m. How much metal (in cm 3) has been used in making the tube?
The volume of a right circular cone is 308 cm 3 and the radius of its base is 7 cm. What is the curved surface area (in cm 2) of the cone? (Take π = \(\frac{22}{7} \) )
The slant height and radius of a right circular cone are in the ratio 29 ∶ 20. If its volume is 4838.4 π cm 3, then its radius is:
Six cubes, each of edge 2 cm, are joined end to end. What is the total surface area of the resulting cuboid in cm 2?
How many metres of 2-m-wide cloth will be required to make a conical tent with a diameter of the base as 14 m and slant height as 9 m? (ignore wastage)
The radius of a right circular cylinder is four times of its height. If the height of the cylinder is 14 cm, then what is the volume of cylinder?
Find the total surface area of a cube whose volume is 343 m3.
Important Questions from Solid Figures
A cone and a hemisphere have equal bases and volumes. What is the ratio of the height of the cone to the radius of the hemisphere?
If the surface area of a sphere is 64 π cm 2, then the volume of the sphere is:
Find the surface area of a sphere of diameter 21 cm. (Use π = \(\frac{{22}}{7}\) )
A cube is 7 cm of an edge and another cube is 14 cm on an edge. The ratios of their surface areas are
Using three distinct points which of the following shapes cannot be formed?
