Find two numbers such that their mean proportional is 18 and the third proportional to them is 144

Options

A. 6 and 42
B. 9 and 36
C. 3 and 18
D. 6 and 12

Answer

To solve the problem of finding two numbers such that their mean proportional is 18 and their third proportional is 144, we can follow these steps:

### Step 1: Define the two numbers
Let the two numbers be \( x \) and \( y \).

### Step 2: Use the mean proportional
The mean proportional between two numbers \( x \) and \( y \) is given by the formula:
\[ \sqrt{xy} = 18 \]
Squaring both sides, we have:
\[ xy = 18^2 = 324 \]

### Step 3: Use the third proportional
The third proportional to \( x \) and \( y \) is given by the formula:
\[ \frac{y}{x} = \frac{144}{y} \]
This can be rearranged to:
\[ y^2 = 144x \]

### Step 4: Substitute for \( x \)
From the first equation \( xy = 324 \), we can express \( x \) in terms of \( y \):
\[ x = \frac{324}{y} \]

### Step 5: Substitute \( x \) into the third proportional equation
Substituting \( x \) into the third proportional equation:
\[ y^2 = 144 \left(\frac{324}{y}\right) \]
Multiplying both sides by \( y \) to eliminate the fraction:
\[ y^3 = 144 \times 324 \]

### Step 6: Calculate \( 144 \times 324 \)
Calculating \( 144 \times 324 \):
\[ 144 \times 324 = 46656 \]
Thus, we have:
\[ y^3 = 46656 \]

### Step 7: Find \( y \)
Taking the cube root of both sides:
\[ y = \sqrt[3]{46656} \]
Calculating the cube root:
\[ y = 36 \]

### Step 8: Find \( x \)
Now substitute \( y \) back into the equation for \( x \):
\[ x = \frac{324}{y} = \frac{324}{36} = 9 \]

### Conclusion
The two numbers are \( x = 9 \) and \( y = 36 \).