The surface areas of two spheres are in the ratio 1 ∶ 4. What is the ratio of their volumes?
1 ∶ 8
Understanding Sphere Ratios: Surface Area to Volume
This question asks us to find the ratio of the volumes of two spheres given the ratio of their surface areas. To solve this, we need to use the formulas for the surface area and volume of a sphere and understand how ratios relate to dimensions.
Key Formulas for a Sphere
Let \(r\) be the radius of a sphere.
- Surface Area (\(A\)) = \(4\pi r^2\)
- Volume (\(V\)) = \(\frac{4}{3}\pi r^3\)
Step-by-Step Solution
Step 1: Relate the Surface Area Ratio to the Radius Ratio
Let the two spheres be Sphere 1 and Sphere 2, with radii \(r_1\) and \(r_2\) respectively, and surface areas \(A_1\) and \(A_2\).
We are given that the ratio of their surface areas is \(A_1 : A_2 = 1 : 4\).
Using the surface area formula:
\(\frac{A_1}{A_2} = \frac{4\pi r_1^2}{4\pi r_2^2}\)
The \(4\pi\) terms cancel out:
\(\frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} = \left(\frac{r_1}{r_2}\right)^2\)
We are given \(\frac{A_1}{A_2} = \frac{1}{4}\), so:
\(\left(\frac{r_1}{r_2}\right)^2 = \frac{1}{4}\)
To find the ratio of the radii \(\frac{r_1}{r_2}\), we take the square root of both sides:
\(\frac{r_1}{r_2} = \sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}\)
So, the ratio of the radii of the two spheres is \(r_1 : r_2 = 1 : 2\). This means if the radius of the first sphere is \(r\), the radius of the second sphere is \(2r\).
Step 2: Relate the Radius Ratio to the Volume Ratio
Now, let \(V_1\) and \(V_2\) be the volumes of Sphere 1 and Sphere 2 respectively.
Using the volume formula:
\(\frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3}\)
The \(\frac{4}{3}\pi\) terms cancel out:
\(\frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} = \left(\frac{r_1}{r_2}\right)^3\)
We found that \(\frac{r_1}{r_2} = \frac{1}{2}\). Substitute this value into the volume ratio formula:
\(\frac{V_1}{V_2} = \left(\frac{1}{2}\right)^3 = \frac{1^3}{2^3} = \frac{1}{8}\)
Thus, the ratio of the volumes of the two spheres is \(V_1 : V_2 = 1 : 8\).
Summary of Ratios
| Quantity | Ratio (Sphere 1 : Sphere 2) |
|---|---|
| Radius (linear dimension) | 1 : 2 |
| Surface Area (square of linear dimension) | \(1^2 : 2^2\) = 1 : 4 (Given) |
| Volume (cube of linear dimension) | \(1^3 : 2^3\) = 1 : 8 (Calculated) |
The ratio of volumes is \(1 : 8\).
Revision Table: Sphere Formulas and Ratios
| Property | Formula (radius \(r\)) | Ratio between two spheres (ratio of radii \(r_1:r_2 = k:1\)) |
|---|---|---|
| Radius | \(r\) | \(k:1\) |
| Surface Area | \(4\pi r^2\) | \(k^2:1\) |
| Volume | \(\frac{4}{3}\pi r^3\) | \(k^3:1\) |
This table illustrates the general principle: if the ratio of corresponding linear dimensions (like radius) is \(k:1\), then the ratio of areas is \(k^2:1\), and the ratio of volumes is \(k^3:1\). In our problem, the area ratio is \(1:4\), meaning \(k^2=1\) and \(1=4\) is incorrect approach. Instead, the ratio is \(1:4\), so \(k^2 = 1/4\), which means \(k = \sqrt{1/4} = 1/2\). So the ratio of radii is \(1/2 : 1\), which simplifies to \(1:2\). The ratio of volumes is then \((1/2)^3 : 1^3 = 1/8 : 1\), which is \(1:8\). This confirms our step-by-step calculation.
Additional Information: Scaling and Dimensions
This concept is not limited to spheres. It applies to any similar 3D shapes. If two similar shapes have corresponding linear dimensions in the ratio \(a:b\), then:
- Their corresponding areas (like surface area, base area) are in the ratio \(a^2:b^2\).
- Their volumes are in the ratio \(a^3:b^3\).
In this specific problem, the radii are the linear dimensions of the spheres. Since spheres are always similar to each other, this scaling principle directly applies.
The given surface area ratio of \(1:4\) corresponds to the \(a^2:b^2\) ratio. Taking the square root gives the linear dimension ratio \(a:b = \sqrt{1}:\sqrt{4} = 1:2\). Then, the volume ratio is \((a:b)^3 = 1^3:2^3 = 1:8\).
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