x and y are two numbers such that their mean proportion Is 9 and third proportion is 243. What are the values of x and y?

Options

A. 3 and 9
B. 3 and 27
C. 6 and 27
D. 6 and 81

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given that their mean proportion is 9 and their third proportion is 243.

### Step-by-Step Solution

1. **Understanding Mean Proportion**:
The mean proportion of two numbers \( x \) and \( y \) is given by the formula:
\[ \sqrt{xy} = 9 \]
Squaring both sides, we have:
\[ xy = 9^2 = 81 \quad \text{(Equation 1)} \]

**Hint**: Recall that the mean proportion of two numbers \( a \) and \( b \) can be expressed as \( \sqrt{ab} \).

2. **Understanding Third Proportion**:
The third proportion of \( x \) and \( y \) is defined as:
\[ \frac{y^2}{x} = 243 \quad \text{(Equation 2)} \]

**Hint**: The third proportion relates to the ratio of the squares of the two numbers.

3. **Expressing \( y^2 \)**:
From Equation 2, we can express \( y^2 \) in terms of \( x \):
\[ y^2 = 243x \]

**Hint**: Rearranging equations can help isolate variables.

4. **Substituting \( y^2 \) into Equation 1**:
We know from Equation 1 that \( xy = 81 \). We can express \( y \) in terms of \( x \):
\[ y = \frac{81}{x} \]
Now substituting this into the expression for \( y^2 \):
\[ \left(\frac{81}{x}\right)^2 = 243x \]
Simplifying this:
\[ \frac{6561}{x^2} = 243x \]

**Hint**: When substituting, ensure to square the entire expression correctly.

5. **Cross Multiplying**:
Cross multiplying gives:
\[ 6561 = 243x^3 \]
Dividing both sides by 243:
\[ x^3 = \frac{6561}{243} \]

**Hint**: Simplifying fractions can often lead to easier calculations.

6. **Calculating \( x^3 \)**:
We can simplify \( \frac{6561}{243} \):
\[ 6561 = 3^8 \quad \text{and} \quad 243 = 3^5 \]
Thus:
\[ x^3 = 3^{8-5} = 3^3 \]
Taking the cube root:
\[ x = 3 \]

**Hint**: Remember that taking roots can simplify your calculations.

7. **Finding \( y \)**:
Now substituting \( x = 3 \) back into the equation \( xy = 81 \):
\[ 3y = 81 \implies y = \frac{81}{3} = 27 \]

**Hint**: Once you find one variable, substitute it back to find the other.

### Final Values
Thus, the values of \( x \) and \( y \) are:
\[ x = 3, \quad y = 27 \]

**Final Answer**: \( x = 3 \) and \( y = 27 \).