The ratio of radii of two hemispheres with surface areas in the ratio 25:49 is 5:7.
Understanding the relationship between the surface area and radius of a hemisphere is key to solving this problem. The surface area of a hemisphere comprises two parts: the curved surface and its flat circular base.
Understanding Hemisphere Surface Area
A hemisphere is half of a sphere.
- Curved Surface Area: For a sphere with radius $r$, its surface area is $4\pi r^2$. Therefore, the curved surface area of a hemisphere is half of that, which is $2\pi r^2$.
- Base Area: The base of a hemisphere is a circle with radius $r$. Its area is $\pi r^2$.
Combining these, the total surface area (S) of a hemisphere is:
$S = (\text{Curved Surface Area}) + (\text{Base Area})$
$S = 2\pi r^2 + \pi r^2$
$S = 3\pi r^2$
This formula shows that the surface area of a hemisphere is directly proportional to the square of its radius ($r^2$).
Calculating the Ratio of Radii
Given that the surface areas of two hemispheres are in the ratio 25:49, we can denote them as $S_1$ and $S_2$, with corresponding radii $r_1$ and $r_2$.
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Set up the ratio of surface areas:
$\frac{S_1}{S_2} = \frac{25}{49}$ -
Substitute the formula for surface area:
Since $S = 3\pi r^2$, we have:
$\frac{3\pi r_1^2}{3\pi r_2^2} = \frac{25}{49}$ -
Simplify the equation:
The $3\pi$ terms cancel out:
$\frac{r_1^2}{r_2^2} = \frac{25}{49}$ -
Solve for the ratio of radii:
To find the ratio of radii, we take the square root of both sides of the equation:
$\sqrt{\frac{r_1^2}{r_2^2}} = \sqrt{\frac{25}{49}}$
$\frac{r_1}{r_2} = \frac{\sqrt{25}}{\sqrt{49}}$
$\frac{r_1}{r_2} = \frac{5}{7}$
Therefore, the ratio of radii is 5:7. This demonstrates that when surface areas are in a certain ratio, their corresponding radii are in the ratio of the square roots of those numbers.
Summary Table
| Property | Hemisphere 1 | Hemisphere 2 | Ratio (H1:H2) |
|---|---|---|---|
| Surface Area (S) | $S_1$ | $S_2$ | 25:49 |
| Radius (r) | $r_1$ | $r_2$ | 5:7 |
| Squared Radius ($r^2$) | $r_1^2$ | $r_2^2$ | 25:49 |
Key Insights
- The relationship between the surface area of a hemisphere and its radius is quadratic ($S \propto r^2$).
- When comparing two hemispheres, the constant $3\pi$ cancels out in the ratio, simplifying the calculation significantly.
- This principle applies to any two-dimensional measure of similar shapes in 3D (like surface area) compared to their linear dimensions (like radius). For spheres, the total surface area formula $4\pi r^2$ would lead to the same ratio of radii. You can learn more about the surface area of a hemisphere and other geometric solids from reliable mathematical resources.
