If the radius of a sphere is 3/4 of the radius of a hemisphere, then what will be the ratio of the volumes of sphere and hemisphere?
27 ∶ 32
Understanding the Problem: Sphere and Hemisphere Volumes
The question asks us to find the ratio of the volume of a sphere to the volume of a hemisphere, given a specific relationship between their radii. We are told that the radius of the sphere is $\frac{3}{4}$ of the radius of the hemisphere.
Formulas for Volumes
To solve this problem, we need the standard formulas for the volume of a sphere and a hemisphere:
- The volume of a sphere with radius $r$ is given by: $$V_{\text{sphere}} = \frac{4}{3} \pi r^3$$
- The volume of a hemisphere with radius $r$ is half the volume of a sphere with the same radius: $$V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$$
Relating the Radii and Calculating the Ratio
Let $r_s$ be the radius of the sphere and $r_h$ be the radius of the hemisphere. According to the question, the relationship between their radii is:
$$r_s = \frac{3}{4} r_h$$Now, let's write the volume of the sphere in terms of $r_h$ using this relationship:
$$V_s = \frac{4}{3} \pi (r_s)^3 = \frac{4}{3} \pi \left(\frac{3}{4} r_h\right)^3$$Let's expand the term $\left(\frac{3}{4} r_h\right)^3$:
$$\left(\frac{3}{4} r_h\right)^3 = \frac{3^3}{4^3} (r_h)^3 = \frac{27}{64} r_h^3$$Substitute this back into the volume of the sphere formula:
$$V_s = \frac{4}{3} \pi \left(\frac{27}{64} r_h^3\right) = \frac{4}{3} \times \frac{27}{64} \pi r_h^3$$Simplify the numerical part:
$$V_s = \frac{4 \times 27}{3 \times 64} \pi r_h^3 = \frac{108}{192} \pi r_h^3$$We can simplify the fraction $\frac{108}{192}$ by dividing both numerator and denominator by their greatest common divisor. Let's divide by 12:
$$\frac{108 \div 12}{192 \div 12} = \frac{9}{16}$$So, the volume of the sphere can be written as:
$$V_s = \frac{9}{16} \pi r_h^3$$The volume of the hemisphere with radius $r_h$ is:
$$V_h = \frac{2}{3} \pi r_h^3$$Now, we need to find the ratio of the volume of the sphere to the volume of the hemisphere, i.e., $\frac{V_s}{V_h}$:
$$\frac{V_s}{V_h} = \frac{\frac{9}{16} \pi r_h^3}{\frac{2}{3} \pi r_h^3}$$Cancel out the common terms $\pi$ and $r_h^3$:
$$\frac{V_s}{V_h} = \frac{\frac{9}{16}}{\frac{2}{3}}$$To divide by a fraction, we multiply by its reciprocal:
$$\frac{V_s}{V_h} = \frac{9}{16} \times \frac{3}{2} = \frac{9 \times 3}{16 \times 2} = \frac{27}{32}$$The ratio of the volumes of the sphere and the hemisphere is $27:32$.
Summary of Steps
- Identify the formulas for the volume of a sphere and a hemisphere.
- Use the given relationship between the radii of the sphere and the hemisphere.
- Express the volume of the sphere in terms of the hemisphere's radius.
- Calculate the ratio of the volume of the sphere to the volume of the hemisphere.
Result
The ratio of the volume of the sphere to the volume of the hemisphere is $27:32$.
Revision Table: Sphere and Hemisphere Volume Ratio
| Shape | Radius | Volume Formula | Volume (in terms of $r_h$) |
|---|---|---|---|
| Sphere | $r_s = \frac{3}{4} r_h$ | $V_s = \frac{4}{3} \pi r_s^3$ | $V_s = \frac{4}{3} \pi (\frac{3}{4} r_h)^3 = \frac{27}{48} \pi r_h^3 = \frac{9}{16} \pi r_h^3$ |
| Hemisphere | $r_h$ | $V_h = \frac{2}{3} \pi r_h^3$ | $V_h = \frac{2}{3} \pi r_h^3$ |
| Ratio $V_s : V_h$ | $\frac{V_s}{V_h} = \frac{\frac{9}{16} \pi r_h^3}{\frac{2}{3} \pi r_h^3} = \frac{9}{16} \times \frac{3}{2} = \frac{27}{32}$ | ||
Additional Information: Sphere and Hemisphere Geometry
Spheres and hemispheres are fundamental 3D geometric shapes with various properties related to surface area and volume. Understanding their formulas is crucial for many geometry problems.
- Sphere: A perfectly round geometrical object in three-dimensional space that is the surface of a perfectly round ball. All points on the surface are equidistant from the center.
- Hemisphere: Half of a sphere, created by cutting a sphere through its center plane. It includes the curved surface and a flat circular base.
- Surface Area: The surface area formulas are different from volume. The surface area of a sphere is $4 \pi r^2$. The total surface area of a hemisphere includes the curved part ($2 \pi r^2$) and the flat base ($\pi r^2$), totaling $3 \pi r^2$. The curved surface area of a hemisphere is $2 \pi r^2$.
- Ratio Dependence: Notice how the $\pi$ and $r_h^3$ terms cancel out in the ratio calculation. This shows that the ratio of the volumes only depends on the numerical relationship between the radii, not their actual values.
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