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

Options

A. `a^(3//2):c^(3//2)`
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 \(c\). Let's go through the solution step by step.

### Step 1: Understand the Mean Proportional
Since \(b\) is the mean proportional of \(a\) and \(c\), we can express this relationship mathematically:
\[ b^2 = ac \]
From this, we can find \(b\):
\[ b = \sqrt{ac} \]

### Step 2: Express \(a - b\) and \(b - c\)
Now, we need to express \(a - b\) and \(b - c\) using the value of \(b\):
\[ a - b = a - \sqrt{ac} \]
\[ b - c = \sqrt{ac} - c \]

### Step 3: Write the Ratio
We need to find the ratio:
\[ \frac{(a - b)^3}{(b - c)^3} \]
Substituting the expressions for \(a - b\) and \(b - c\):
\[ \frac{(a - \sqrt{ac})^3}{(\sqrt{ac} - c)^3} \]

### Step 4: Factor Out Common Terms
To simplify the expression, we can factor out \(\sqrt{a}\) from the numerator and \(\sqrt{c}\) from the denominator:
1. In the numerator:
\[ a - \sqrt{ac} = \sqrt{a}(\sqrt{a} - \sqrt{c}) \]
So,
\[ (a - b)^3 = \left(\sqrt{a}(\sqrt{a} - \sqrt{c})\right)^3 = a^{3/2}(\sqrt{a} - \sqrt{c})^3 \]

2. In the denominator:
\[ \sqrt{ac} - c = \sqrt{c}(\sqrt{a} - \sqrt{c}) \]
So,
\[ (b - c)^3 = \left(\sqrt{c}(\sqrt{a} - \sqrt{c})\right)^3 = c^{3/2}(\sqrt{a} - \sqrt{c})^3 \]

### Step 5: Simplify the Ratio
Now we can substitute these back into our ratio:
\[ \frac{a^{3/2}(\sqrt{a} - \sqrt{c})^3}{c^{3/2}(\sqrt{a} - \sqrt{c})^3} \]
The \((\sqrt{a} - \sqrt{c})^3\) terms cancel out:
\[ \frac{a^{3/2}}{c^{3/2}} \]

### Final Result
Thus, the final result for the ratio \((a-b)^3 : (b-c)^3\) is:
\[ \frac{a^{3/2}}{c^{3/2}} \quad \text{or} \quad a^{3/2} : c^{3/2} \]