a and b are two numbers such that their mean proportion is 6 and third proportion is 48. What is the value of a and b?

Options

A. 6,6
B. 3,6
C. 6,12
D. 3,12

Answer

To solve the problem step by step, we will use the definitions of mean proportion and third proportion.

### Step 1: Understand the Mean Proportion
The mean proportion of two numbers \( a \) and \( b \) is defined as:
\[ \text{Mean Proportion} = \sqrt{a \cdot b} \]
According to the problem, the mean proportion is given as 6. Therefore, we can write:
\[ \sqrt{a \cdot b} = 6 \]

### Step 2: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ a \cdot b = 6^2 \]
This simplifies to:
\[ a \cdot b = 36 \quad \text{(Equation 1)} \]

### Step 3: Understand the Third Proportion
The third proportion of two numbers \( a \) and \( b \) is defined as:
\[ \text{Third Proportion} = \frac{b^2}{a} \]
According to the problem, the third proportion is given as 48. Therefore, we can write:
\[ \frac{b^2}{a} = 48 \]

### Step 4: Rearranging the Third Proportion Equation
From the third proportion equation, we can express \( b^2 \) in terms of \( a \):
\[ b^2 = 48a \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
From Equation 1, we have \( b = \frac{36}{a} \). We can substitute this value of \( b \) into Equation 2:
\[ \left(\frac{36}{a}\right)^2 = 48a \]

### Step 6: Simplify the Equation
Squaring \( \frac{36}{a} \) gives us:
\[ \frac{1296}{a^2} = 48a \]
Now, multiply both sides by \( a^2 \) to eliminate the fraction:
\[ 1296 = 48a^3 \]

### Step 7: Solve for \( a^3 \)
Now, divide both sides by 48:
\[ a^3 = \frac{1296}{48} \]
Calculating the right side gives:
\[ a^3 = 27 \]

### Step 8: Find the Value of \( a \)
Taking the cube root of both sides:
\[ a = 3 \]

### Step 9: Find the Value of \( b \)
Now, substitute \( a = 3 \) back into Equation 1 to find \( b \):
\[ 3 \cdot b = 36 \]
Dividing both sides by 3 gives:
\[ b = 12 \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 3, \quad b = 12 \]