`{:("Column A","The arithmetic mean (average) of the number a and b is 17. The geometric mean of the numbers a and b is 8. The geometric mean of two numbers is defined to be the square root of their product","Column B"),(a,,b):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to analyze the information given about the numbers \( a \) and \( b \).

### Step 1: Understand the given means
We know:
- The arithmetic mean (AM) of \( a \) and \( b \) is 17.
- The geometric mean (GM) of \( a \) and \( b \) is 8.

### Step 2: Set up the equations
From the definition of the arithmetic mean:
\[ \text{AM} = \frac{a + b}{2} = 17 \]
Multiplying both sides by 2 gives:
\[ a + b = 34 \quad \text{(Equation 1)} \]

From the definition of the geometric mean:
\[ \text{GM} = \sqrt{ab} = 8 \]
Squaring both sides gives:
\[ ab = 64 \quad \text{(Equation 2)} \]

### Step 3: Use the equations to find \( a \) and \( b \)
We now have two equations:
1. \( a + b = 34 \)
2. \( ab = 64 \)

These equations represent a system of equations that can be solved using the quadratic formula. We can express \( a \) and \( b \) as the roots of the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \]
Substituting the values from our equations:
\[ x^2 - 34x + 64 = 0 \]

### Step 4: Calculate the discriminant
To find the roots, we first calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-34)^2 - 4 \cdot 1 \cdot 64 = 1156 - 256 = 900 \]

### Step 5: Find the roots
Since the discriminant is positive, we can find the roots using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{34 \pm \sqrt{900}}{2} \]
Calculating \( \sqrt{900} = 30 \):
\[ x = \frac{34 \pm 30}{2} \]
This gives us two possible values:
1. \( x = \frac{64}{2} = 32 \)
2. \( x = \frac{4}{2} = 2 \)

Thus, the two numbers are \( a = 32 \) and \( b = 2 \) (or vice versa).

### Step 6: Compare \( a \) and \( b \)
Now we can compare the two values:
- \( a = 32 \)
- \( b = 2 \)

Clearly, \( a > b \).

### Conclusion
Since we have determined the values of \( a \) and \( b \), we conclude that the value in Column A (which is \( a \)) is greater than the value in Column B (which is \( b \)). Therefore, the answer is that Column A is greater.