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?
2 ∶ 1
Solving the Cone and Hemisphere Volume Ratio Problem
This problem asks us to find the ratio of the height of a cone to the radius of a hemisphere, given that they have equal bases and equal volumes.
Understanding the Problem Conditions
- Equal Bases: The base of a cone is a circle, and the base of a hemisphere is also a circle (the great circle). Having equal bases means their base areas are equal. The area of a circle is given by \(\pi r^2\). If the radius of the cone's base is \(r_c\) and the radius of the hemisphere's base is \(r_h\), then \(\pi r_c^2 = \pi r_h^2\). This implies \(r_c^2 = r_h^2\), and since radii are positive, we have \(r_c = r_h\). Let's denote this common radius by \(r\). So, the radius of the cone's base is \(r\), and the radius of the hemisphere is also \(r\).
- Equal Volumes: The volume of the cone is equal to the volume of the hemisphere.
Formulas for Volume
We need the formulas for the volume of a cone and a hemisphere:
- Volume of a cone (\(V_c\)) with radius \(r_c\) and height \(h_c\): \(V_c = \frac{1}{3} \pi r_c^2 h_c\)
- Volume of a hemisphere (\(V_h\)) with radius \(r_h\): \(V_h = \frac{2}{3} \pi r_h^3\)
Setting Up the Equation
We are given that the volumes are equal, \(V_c = V_h\). Using the common radius \(r\) (since \(r_c = r_h = r\)), we can write:
\(\frac{1}{3} \pi r^2 h_c = \frac{2}{3} \pi r^3\)
Solving for the Ratio of Height to Radius
Now, we need to solve this equation for the ratio \(\frac{h_c}{r}\). Let's simplify the equation:
- Cancel out the common factors on both sides. Both sides have \(\frac{1}{3}\pi\) and \(r^2\) (assuming \(r \neq 0\), which is true for shapes with volume).
Dividing both sides by \(\frac{1}{3}\pi r^2\):
\(\frac{\frac{1}{3} \pi r^2 h_c}{\frac{1}{3} \pi r^2} = \frac{\frac{2}{3} \pi r^3}{\frac{1}{3} \pi r^2}\)
\(h_c = \frac{2 r^3}{r^2}\)
\(h_c = 2r\)
The question asks for the ratio of the height of the cone (\(h_c\)) to the radius of the hemisphere. The radius of the hemisphere is \(r_h\), which we established is equal to \(r\). So, we need to find the ratio \(\frac{h_c}{r_h} = \frac{h_c}{r}\).
From our equation \(h_c = 2r\), we can rearrange it to find the ratio:
\(\frac{h_c}{r} = 2\)
This can be written as a ratio \(2:1\).
Conclusion
The ratio of the height of the cone to the radius of the hemisphere, when they have equal bases and equal volumes, is \(2:1\).
| Shape | Radius (Base/Hemisphere) | Height (Cone) | Volume Formula |
|---|---|---|---|
| Cone | \(r_c = r\) | \(h_c\) | \(V_c = \frac{1}{3} \pi r_c^2 h_c = \frac{1}{3} \pi r^2 h_c\) |
| Hemisphere | \(r_h = r\) | N/A | \(V_h = \frac{2}{3} \pi r_h^3 = \frac{2}{3} \pi r^3\) |
Given \(V_c = V_h\):
\(\frac{1}{3} \pi r^2 h_c = \frac{2}{3} \pi r^3\)
\(h_c = 2r\)
Ratio \(\frac{h_c}{r_h} = \frac{h_c}{r} = \frac{2r}{r} = \frac{2}{1}\)
Revision Table: Cone and Hemisphere Properties
| Property | Cone | Hemisphere |
|---|---|---|
| Base Shape | Circle | Circle |
| Base Area (radius \(r\)) | \(\pi r^2\) | \(\pi r^2\) |
| Volume (radius \(r\), cone height \(h\)) | \(\frac{1}{3} \pi r^2 h\) | \(\frac{2}{3} \pi r^3\) |
| Equal Bases Condition | Base radii are equal (\(r_c = r_h\)) | |
| Equal Volumes Condition | \(V_{cone} = V_{hemisphere}\) | |
Additional Information: Solids of Revolution and Volumes
The cone and hemisphere are examples of common three-dimensional geometric shapes. Their volumes are derived using calculus (integration) by considering them as solids of revolution or through other geometric methods.
- A cone can be formed by rotating a right-angled triangle about one of its perpendicular sides (the height).
- A sphere (of which a hemisphere is half) can be formed by rotating a circle about its diameter.
Understanding these fundamental shapes and their volume formulas is crucial for solving many geometry and mensuration problems. The principle of equating volumes or surface areas based on given conditions is a common technique in these types of questions.
Similar Questions
The volume of a hemisphere is 155232 cm 3. What is the radius of the hemisphere?
The radius and height of a right circular cone are in the ratio 3 : 7. If the volume of the cone is 528 cm 3, then what is the height of the cone? \(\left( {{\rm{Take}}\,\,{\rm{\pi }}\,{\rm{ = }}\frac{{22}}{7}} \right)\)
The length, breadth and height of a cuboid are in the ratio 27 : 8 : 1. The cuboid is melted and recast into a cube. If p is the surface area of the cuboid and q is the surface area of the cube, then what is p/q equal to?
If the radius of a sphere is rational, then which of the following is/are correct?
1. Its surface area is rational.
2. Its volume is rational.Select the correct answer using the code given below:
The surface area of a cube is equal to that of a sphere. If x is the volume of the cube and y is the volume of the sphere, then what is x 2 ∶ y 2 equal to?
The areas of two circular fields are in the ratio 16 ∶ 49. If the radius of the bigger field is 14 m then what is the radius of the smaller field?
If the radius of a right circular cone is increased by p% without increasing its height, then what is the percentage increase in the volume of the cone?
If the surface area of a sphere is reduced to one-ninth of the area, its radius reduces to
A solid cube is cut into two cuboids of equal volume. What is the ratio of total surface area of the given cube to that of one of the cuboids ?
Two times the total surface area of a solid right circular cylinder is three times its curved surface area. If h is the height and r is the radius of the base of the cylinder, then which one of the following is correct ?
Important Questions from Solid Figures
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?
Two cubes each of edge 12 cm are joined end to end. Find the surface area of the resulting cuboid.
The areas of three adjacent faces of a cuboid are x, y, z. If the volume is V, then V 2will be equal to:
The base radius of a cylinder is 14 cm. If curved surface area of the cylinder is three-fifth of the total surface area, then what will be the volume (in cu. cm) of the cylinder? (Use π = \(\frac{{22}}{7}\) )
