If the surface area of a sphere is reduced to one-ninth of the area, its radius reduces to
One-third
Understanding Sphere Surface Area and Radius
The question asks us to find out how the radius of a sphere changes when its surface area is reduced to one-ninth of its original size. To solve this, we need to know the formula for the surface area of a sphere and understand the relationship between the area and the radius.
Formula for Sphere Surface Area
The surface area (\(A\)) of a sphere with radius (\(r\)) is given by the formula:
\( A = 4\pi r^2 \)
This formula shows that the surface area is directly proportional to the square of the radius.
Relating Original and Reduced Surface Areas
Let's denote the original surface area as \(A_1\) and the original radius as \(r_1\). So, \(A_1 = 4\pi r_1^2\).
Let the new surface area after reduction be \(A_2\) and the new radius be \(r_2\). So, \(A_2 = 4\pi r_2^2\).
According to the problem statement, the new surface area \(A_2\) is one-ninth of the original surface area \(A_1\). We can write this relationship as:
\( A_2 = \frac{1}{9} A_1 \)
Solving for the New Radius
Now, we substitute the formulas for \(A_1\) and \(A_2\) into the equation \(A_2 = \frac{1}{9} A_1\):
\( 4\pi r_2^2 = \frac{1}{9} (4\pi r_1^2) \)
We can cancel the common term \(4\pi\) from both sides of the equation:
\( r_2^2 = \frac{1}{9} r_1^2 \)
To find the relationship between the radii, we take the square root of both sides of the equation:
\( \sqrt{r_2^2} = \sqrt{\frac{1}{9} r_1^2} \)
\( r_2 = \sqrt{\frac{1}{9}} \times \sqrt{r_1^2} \)
Since \(\sqrt{\frac{1}{9}} = \frac{1}{3}\) and \(\sqrt{r_1^2} = r_1\) (radius is positive), we get:
\( r_2 = \frac{1}{3} r_1 \)
This result shows that the new radius \(r_2\) is one-third of the original radius \(r_1\).
Conclusion on Radius Reduction
When the surface area of a sphere is reduced to one-ninth, its radius reduces to one-third of the original radius. This is because the surface area depends on the square of the radius. If the area scales by a factor of \(k\), the radius scales by a factor of \(\sqrt{k}\). In this case, \(k = \frac{1}{9}\), so the radius scales by \(\sqrt{\frac{1}{9}} = \frac{1}{3}\).
Analyzing the Options
- One-fourth: This would imply the surface area ratio was \((1/4)^2 = 1/16\), which is incorrect.
- One-third: This matches our calculation, as the surface area ratio is \((1/3)^2 = 1/9\).
- One-fifth: This would imply the surface area ratio was \((1/5)^2 = 1/25\), which is incorrect.
- One-ninth: This would imply the surface area ratio was \((1/9)^2 = 1/81\), which is incorrect.
| Quantity | Original (\(1\)) | New (\(2\)) | Relationship |
|---|---|---|---|
| Surface Area (\(A\)) | \(A_1\) | \(A_2\) | \(A_2 = \frac{1}{9} A_1\) |
| Radius (\(r\)) | \(r_1\) | \(r_2\) | \(r_2 = \frac{1}{3} r_1\) |
| Formula Relationship | \(A_1 = 4\pi r_1^2\) | \(A_2 = 4\pi r_2^2\) | \(4\pi r_2^2 = \frac{1}{9} (4\pi r_1^2)\) |
Revision Table: Sphere Geometry
| Property | Formula (Radius \(r\)) |
|---|---|
| Surface Area | \(A = 4\pi r^2\) |
| Volume | \(V = \frac{4}{3}\pi r^3\) |
Understanding how dimensions (like radius) affect area and volume is crucial in geometry. For shapes where area depends on the square of a linear dimension, if the area changes by a factor \(k\), the linear dimension changes by \(\sqrt{k}\).
Additional Information: Scaling in Geometry
When you scale a geometric shape, the way its properties change depends on the dimension of the property:
- Linear dimensions (like radius, diameter, side length): If you scale all linear dimensions by a factor \(s\), the new linear dimensions are \(s\) times the original.
- Area (like surface area): If you scale all linear dimensions by a factor \(s\), the new area is \(s^2\) times the original area.
- Volume: If you scale all linear dimensions by a factor \(s\), the new volume is \(s^3\) times the original volume.
In our problem, the surface area is scaled by a factor of \(k = 1/9\). Since area scales as the square of the radius \(A \propto r^2\), the scaling factor for the radius (\(s\)) must satisfy \(s^2 = k\). So, \(s^2 = 1/9\), which means \(s = \sqrt{1/9} = 1/3\). This confirms that the radius is scaled by a factor of one-third.
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