The mean proportional between 8 and 97 is:

Options

A. 16
B. 53
C. 112
D. 28

Answer

To find the mean proportional between 8 and 97, we can follow these steps:

### Step 1: Understand the concept of mean proportional
The mean proportional \( A \) between two numbers \( a \) and \( b \) is defined such that the ratio \( \frac{a}{A} = \frac{A}{b} \). This means that \( A \) is the geometric mean of \( a \) and \( b \).

### Step 2: Set up the equation
For our specific values, we have:
- \( a = 8 \)
- \( b = 97 \)

Thus, we can write the equation:
\[ \frac{8}{A} = \frac{A}{97} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 8 \cdot 97 = A \cdot A \]
This simplifies to:
\[ 8 \times 97 = A^2 \]

### Step 4: Calculate \( 8 \times 97 \)
Now, we calculate \( 8 \times 97 \):
\[ 8 \times 97 = 776 \]
So, we have:
\[ A^2 = 776 \]

### Step 5: Solve for \( A \)
To find \( A \), we take the square root of both sides:
\[ A = \sqrt{776} \]

### Step 6: Calculate \( \sqrt{776} \)
Now we can calculate \( \sqrt{776} \). Approximating this gives:
\[ \sqrt{776} \approx 27.856 \]
For practical purposes, we can round this to:
\[ A \approx 28 \]

### Conclusion
Thus, the mean proportional between 8 and 97 is approximately \( 28 \).

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