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.
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