A and B are two numbers. 6 times of square of B is 540 more than the square of A. If the respective ratio between A and B is ` 3 : 2`, what is the value of B ?

Options

A. 10
B. 12
C. 16
D. 8

Explanation

To solve the problem, we will follow these steps:

### Step 1: Set up the equations based on the given information.
We know that the ratio of A to B is 3:2. We can express A in terms of B:
\[ A = \frac{3}{2}B \]

### Step 2: Write the equation based on the relationship given in the problem.
The problem states that six times the square of B is 540 more than the square of A:
\[ 6B^2 = A^2 + 540 \]

### Step 3: Substitute the expression for A into the equation.
Substituting \( A = \frac{3}{2}B \) into the equation:
\[ 6B^2 = \left(\frac{3}{2}B\right)^2 + 540 \]

### Step 4: Simplify the equation.
Calculating \( \left(\frac{3}{2}B\right)^2 \):
\[ \left(\frac{3}{2}B\right)^2 = \frac{9}{4}B^2 \]
Now substitute this back into the equation:
\[ 6B^2 = \frac{9}{4}B^2 + 540 \]

### Step 5: Eliminate fractions by multiplying the entire equation by 4.
To eliminate the fraction, multiply the entire equation by 4:
\[ 4 \cdot 6B^2 = 4 \cdot \frac{9}{4}B^2 + 4 \cdot 540 \]
This simplifies to:
\[ 24B^2 = 9B^2 + 2160 \]

### Step 6: Rearrange the equation.
Now, rearranging gives:
\[ 24B^2 - 9B^2 = 2160 \]
\[ 15B^2 = 2160 \]

### Step 7: Solve for \( B^2 \).
Dividing both sides by 15:
\[ B^2 = \frac{2160}{15} \]
Calculating this gives:
\[ B^2 = 144 \]

### Step 8: Find the value of B.
Taking the square root of both sides:
\[ B = \sqrt{144} \]
Thus:
\[ B = 12 \]

### Final Answer:
The value of B is 12.

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