The arithmetic mean of two positive numbers is `6` and their geometric mean `G` and harmonic mean `H` satisfy the relation `G^(2)+3H=48`. Then the product of the two numbers is

Options

A. `24`
B. `32`
C. `48`
D. `54`

Explanation

To solve the problem step by step, we will use the definitions of arithmetic mean, geometric mean, and harmonic mean.

**Step 1: Define the two positive numbers.**
Let the two positive numbers be \( A \) and \( B \).

**Step 2: Use the information about the arithmetic mean.**
The arithmetic mean (AM) of \( A \) and \( B \) is given by:
\[ \text{AM} = \frac{A + B}{2} = 6 \]
From this, we can derive:
\[ A + B = 12 \quad \text{(Equation 1)} \]

**Step 3: Define the geometric mean.**
The geometric mean (GM) of \( A \) and \( B \) is given by:
\[ G = \sqrt{AB} \]
Thus, we have:
\[ G^2 = AB \quad \text{(Equation 2)} \]

**Step 4: Define the harmonic mean.**
The harmonic mean (HM) of \( A \) and \( B \) is given by:
\[ H = \frac{2AB}{A + B} \]
Substituting \( A + B = 12 \) from Equation 1, we get:
\[ H = \frac{2AB}{12} = \frac{AB}{6} \quad \text{(Equation 3)} \]

**Step 5: Use the given relation involving G and H.**
We are given the relation:
\[ G^2 + 3H = 48 \]
Substituting \( G^2 = AB \) (from Equation 2) and \( H = \frac{AB}{6} \) (from Equation 3) into this equation:
\[ AB + 3\left(\frac{AB}{6}\right) = 48 \]

**Step 6: Simplify the equation.**
This simplifies to:
\[ AB + \frac{3AB}{6} = 48 \]
\[ AB + \frac{1}{2}AB = 48 \]
Combining the terms gives:
\[ \frac{3AB}{2} = 48 \]

**Step 7: Solve for AB.**
To find \( AB \), multiply both sides by 2:
\[ 3AB = 96 \]
Now, divide by 3:
\[ AB = 32 \]

Thus, the product of the two numbers \( A \) and \( B \) is \( \boxed{32} \).

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