If b is the mean proportional of a and c, then `(a - b)^3 : (b - c)^3` equals

Options

A. `a^(3) : c^(3)`
B. `b^(2) : c^(2)`
C. `a^(2) : c^(2)`
D. `a^(3) : b^(3)`

Answer

To solve the problem, we need to find the ratio \((a - b)^3 : (b - c)^3\) given that \(b\) is the mean proportional of \(a\) and \(c\).

### Step-by-Step Solution:

1. **Understanding Mean Proportional**:
Since \(b\) is the mean proportional of \(a\) and \(c\), we can express this relationship mathematically as:
\[ b = \sqrt{ac} \]

2. **Expressing \(a\) and \(c\) in terms of \(b\)**:
We can also express \(a\) and \(c\) in terms of \(b\). Let's assume:
\[ a = k \cdot b \quad \text{and} \quad c = \frac{b}{k} \]
for some positive constant \(k\).

3. **Finding \(a - b\) and \(b - c\)**:
Now, we can find \(a - b\) and \(b - c\):
\[ a - b = k \cdot b - b = (k - 1)b \]
\[ b - c = b - \frac{b}{k} = b\left(1 - \frac{1}{k}\right) = b\left(\frac{k - 1}{k}\right) \]

4. **Cubing the Differences**:
Next, we need to cube both differences:
\[ (a - b)^3 = ((k - 1)b)^3 = (k - 1)^3 b^3 \]
\[ (b - c)^3 = \left(b\left(\frac{k - 1}{k}\right)\right)^3 = \left(\frac{k - 1}{k}\right)^3 b^3 \]

5. **Setting Up the Ratio**:
Now we can set up the ratio:
\[ (a - b)^3 : (b - c)^3 = (k - 1)^3 b^3 : \left(\frac{k - 1}{k}\right)^3 b^3 \]

6. **Simplifying the Ratio**:
The \(b^3\) terms cancel out:
\[ (k - 1)^3 : \left(\frac{k - 1}{k}\right)^3 = (k - 1)^3 : \frac{(k - 1)^3}{k^3} \]
This simplifies to:
\[ (k - 1)^3 \cdot k^3 : (k - 1)^3 = k^3 : 1 \]

7. **Final Result**:
Therefore, the final result is:
\[ (a - b)^3 : (b - c)^3 = k^3 : 1 \]

### Conclusion:
Since \(k\) is a constant representing the ratio of \(a\) to \(b\), we can conclude that:
\[ \boxed{(a - b)^3 : (b - c)^3 = a^3 : c^3} \]