The volume of a hemisphere is 155232 cm 3. What is the radius of the hemisphere?
42 cm
Finding the Radius of a Hemisphere from Volume
The problem asks us to find the radius of a hemisphere when its volume is given as 155232 cubic centimeters.
A hemisphere is exactly half of a sphere. The formula for the volume of a sphere with radius \(r\) is \(V_{sphere} = \frac{4}{3} \pi r^3\). Therefore, the volume of a hemisphere is half of the volume of a sphere.
The formula for the volume of a hemisphere (\(V_{hemisphere}\)) is:
\[V_{hemisphere} = \frac{1}{2} \times V_{sphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3\]
We are given that the volume of the hemisphere is 155232 cm³. We need to substitute this value into the formula and solve for the radius, \(r\).
\[155232 = \frac{2}{3} \pi r^3\]
To solve for \(r^3\), we can rearrange the equation:
\[r^3 = \frac{155232 \times 3}{2 \pi}\]
Using the approximation \(\pi \approx \frac{22}{7}\), the equation becomes:
\[r^3 = \frac{155232 \times 3}{2 \times \frac{22}{7}}\]
\[r^3 = \frac{155232 \times 3 \times 7}{2 \times 22}\]
\[r^3 = \frac{155232 \times 21}{44}\]
Now, let's perform the calculation:
\[r^3 = \frac{3259872}{44}\]
\[r^3 = 74088\]
To find the radius \(r\), we need to calculate the cube root of 74088.
\[r = \sqrt[3]{74088}\]
Let's test the options provided to see which one gives 74088 when cubed:
- If \(r = 40\) cm, \(r^3 = 40^3 = 40 \times 40 \times 40 = 64000\)
- If \(r = 42\) cm, \(r^3 = 42^3 = 42 \times 42 \times 42 = 1764 \times 42 = 74088\)
- If \(r = 38\) cm, \(r^3 = 38^3 = 38 \times 38 \times 38 = 1444 \times 38 = 54872\)
- If \(r = 36\) cm, \(r^3 = 36^3 = 36 \times 36 \times 36 = 1296 \times 36 = 46656\)
The cube root of 74088 is 42.
Thus, the radius of the hemisphere is 42 cm.
The steps involved were:
- Identify the formula for the volume of a hemisphere.
- Substitute the given volume into the formula.
- Solve the equation for the radius \(r\).
- Calculate the cube root to find the value of \(r\).
Revision Table: Hemisphere Volume Calculation
| Concept | Formula | Given Value | To Find |
|---|---|---|---|
| Volume of Sphere | \(V_{sphere} = \frac{4}{3} \pi r^3\) | N/A | Radius (\(r\)) |
| Volume of Hemisphere | \(V_{hemisphere} = \frac{2}{3} \pi r^3\) | 155232 cm³ | Radius (\(r\)) |
Additional Information: Properties of Hemispheres
A hemisphere is a three-dimensional geometric shape that is half of a sphere. It is formed by cutting a sphere through its exact center.
- Shape: A hemisphere has one flat circular face (the base) and one curved face (the dome).
- Radius: The radius (\(r\)) is the distance from the center of the flat base to any point on the edge of the base or to any point on the curved surface, provided the measurement is along a radial line from the center.
- Surface Area:
- Curved Surface Area: \(A_{curved} = 2 \pi r^2\)
- Area of the Base: \(A_{base} = \pi r^2\)
- Total Surface Area: \(A_{total} = A_{curved} + A_{base} = 2 \pi r^2 + \pi r^2 = 3 \pi r^2\)
- Volume: As discussed, \(V_{hemisphere} = \frac{2}{3} \pi r^3\).
These formulas are essential for solving problems involving hemispheres in geometry andMensuration.
Similar Questions
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?
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?
A lamp shade is in the shape of a part of a cone and its top and bottom ends are circles whose circumferences are respectively 30 cm and 40 cm. The perpendicular distance between the ends is 6 cm. If the cone were to be completed, then how far would its vertex be from the top end?
Three solid lead spheres of radius 6 cm, 8 cm and 10 cm are melted together and recast as a solid sphere. What is the percentage diminution of the surface area as compared to the sum of the surface areas of the three spheres ?
A solid sphere of radius 3 cm is melted to form a hollow cylinder of height 4 cm and external diameter 10 cm. What is the thickness of the cylinder?
What is the radius of the base of the cone ?
The surface areas of two spheres are in the ratio 1 ∶ 4. What is the ratio of their volumes?
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:
If H, C and V are respectively the height, curved surface area and volume of a cone, then what is 3πVH 3+ 9V 2equal to?
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?
