`6` is the mean proportion between two numbers `x` and `y` and `48` is third proportion to `x` and `y`. Find the numbers.

Answer

To solve the problem step by step, we need to find the two numbers \( x \) and \( y \) given that \( 6 \) is the mean proportion between them and \( 48 \) is the third proportion to \( x \) and \( y \).

### Step 1: Mean Proportion Equation
Since \( 6 \) is the mean proportion between \( x \) and \( y \), we can express this relationship mathematically as:
\[ 6 = \sqrt{xy} \]
Squaring both sides gives us:
\[ 36 = xy \quad \text{(Equation 1)} \]

### Step 2: Third Proportion Equation
We are also given that \( 48 \) is the third proportion to \( x \) and \( y \). The relationship for the third proportion can be expressed as:
\[ y^2 = 48x \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 1 into Equation 2
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = \frac{36}{x} \]
Now, substitute this value of \( y \) into Equation 2:
\[ \left(\frac{36}{x}\right)^2 = 48x \]
This simplifies to:
\[ \frac{1296}{x^2} = 48x \]

### Step 4: Clear the Fraction
To eliminate the fraction, multiply both sides by \( x^2 \):
\[ 1296 = 48x^3 \]

### Step 5: Solve for \( x^3 \)
Now, divide both sides by \( 48 \):
\[ x^3 = \frac{1296}{48} \]
Calculating the right side:
\[ x^3 = 27 \]

### Step 6: Find \( x \)
Taking the cube root of both sides gives:
\[ x = 3 \]

### Step 7: Find \( y \)
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ xy = 36 \]
Substituting \( x = 3 \):
\[ 3y = 36 \]
Dividing both sides by \( 3 \):
\[ y = 12 \]

### Final Answer
Thus, the two numbers are:
\[ x = 3 \quad \text{and} \quad y = 12 \]