Average of two nurnbers is 6.5 and square root of their product is 6. What are the numbers?

Options

A. (a) 11 and 2
B. (b) 8 and 5
C. (c) 10 and 3
D. (d) 9 and 4

Explanation

To solve the problem, we need to find two numbers based on the information given about their average and the square root of their product.

### Step 1: Understand the average of two numbers
The average of two numbers \( x \) and \( y \) is given by:
\[ \text{Average} = \frac{x + y}{2} \]
Given that the average is 6.5, we can set up the equation:
\[ \frac{x + y}{2} = 6.5 \]
Multiplying both sides by 2 gives:
\[ x + y = 13 \quad \text{(Equation 1)} \]

### Step 2: Understand the product of the two numbers
We are also given that the square root of their product is 6. This can be expressed as:
\[ \sqrt{xy} = 6 \]
Squaring both sides gives:
\[ xy = 36 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of equations:
1. \( x + y = 13 \)
2. \( xy = 36 \)

We can use these equations to find \( x \) and \( y \). From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 13 - x \]

### Step 4: Substitute into the product equation
Substituting \( y \) into Equation 2 gives:
\[ x(13 - x) = 36 \]
Expanding this, we get:
\[ 13x - x^2 = 36 \]
Rearranging the equation gives:
\[ x^2 - 13x + 36 = 0 \]

### Step 5: Factor the quadratic equation
Now we need to factor the quadratic equation:
\[ x^2 - 13x + 36 = 0 \]
We look for two numbers that multiply to 36 and add up to -13. The numbers -9 and -4 work:
\[ (x - 9)(x - 4) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

### Step 7: Find the corresponding \( y \) values
Using \( y = 13 - x \):
- If \( x = 9 \), then \( y = 13 - 9 = 4 \).
- If \( x = 4 \), then \( y = 13 - 4 = 9 \).

Thus, the two numbers are 9 and 4.

### Final Answer
The two numbers are **4 and 9**.