The geometric mean `G` of two positive numbers is `6`. Their arithmetic mean `A` and harmonic mean `H` satisfy the equation `90A+5H=918`, then `A` may be equal to:

Options

A. (A) `5/2`
B. (B) `10`
C. (C) `5`
D. (D) `1/5`

Explanation

To solve the problem, we need to find the value of the arithmetic mean \( A \) given that the geometric mean \( G \) of two positive numbers is \( 6 \) and that the arithmetic mean \( A \) and harmonic mean \( H \) satisfy the equation \( 90A + 5H = 918 \).

### Step-by-Step Solution:

1. **Understanding Geometric Mean**:
The geometric mean \( G \) of two numbers \( x \) and \( y \) is given by:
\[ G = \sqrt{xy} \]
Given that \( G = 6 \), we can square both sides:
\[ xy = 6^2 = 36 \quad \text{(Equation 1)} \]

2. **Finding Arithmetic Mean**:
The arithmetic mean \( A \) of \( x \) and \( y \) is given by:
\[ A = \frac{x + y}{2} \]
Rearranging gives:
\[ x + y = 2A \quad \text{(Equation 2)} \]

3. **Finding Harmonic Mean**:
The harmonic mean \( H \) of \( x \) and \( y \) is given by:
\[ H = \frac{2xy}{x + y} \]
Substituting \( xy = 36 \) from Equation 1 and \( x + y = 2A \) from Equation 2:
\[ H = \frac{2 \cdot 36}{2A} = \frac{72}{2A} = \frac{36}{A} \quad \text{(Equation 3)} \]

4. **Substituting into the Given Equation**:
We are given the equation:
\[ 90A + 5H = 918 \]
Substituting \( H \) from Equation 3:
\[ 90A + 5 \left( \frac{36}{A} \right) = 918 \]
This simplifies to:
\[ 90A + \frac{180}{A} = 918 \]

5. **Multiplying through by \( A \)**:
To eliminate the fraction, multiply the entire equation by \( A \):
\[ 90A^2 + 180 = 918A \]
Rearranging gives:
\[ 90A^2 - 918A + 180 = 0 \]

6. **Dividing the Equation**:
To simplify, divide the entire equation by \( 90 \):
\[ A^2 - 10.2A + 2 = 0 \]

7. **Using the Quadratic Formula**:
The quadratic formula is:
\[ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -10.2 \), and \( c = 2 \):
\[ A = \frac{10.2 \pm \sqrt{(-10.2)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
Calculating the discriminant:
\[ A = \frac{10.2 \pm \sqrt{104.04 - 8}}{2} \]
\[ A = \frac{10.2 \pm \sqrt{96.04}}{2} \]
\[ A = \frac{10.2 \pm 9.8}{2} \]

8. **Finding the Roots**:
This gives us two possible values for \( A \):
\[ A = \frac{20}{2} = 10 \quad \text{and} \quad A = \frac{0.4}{2} = 0.2 \]

### Conclusion:
Thus, the possible values for \( A \) are \( 10 \) and \( 0.2 \).