Two forces are such that the sum of their magnitudes is 18 N, the resultant is `sqrt(228)` when they are at `120^(@)`. Then the magnitude of the forces are

Options

A. 12N,6N
B. 13N,5N
C. 10N, 9N
D. 16N, 2N

Answer

To solve the problem, we need to find the magnitudes of two forces \( p \) and \( q \) given the following conditions:

1. The sum of their magnitudes is \( p + q = 18 \, \text{N} \).
2. The resultant of the two forces when they act at an angle of \( 120^\circ \) is \( R = \sqrt{228} \, \text{N} \).

### Step 1: Set up the equations
From the problem, we have:
- \( p + q = 18 \) (1)
- The formula for the resultant of two forces is given by:
\[ R = \sqrt{p^2 + q^2 + 2pq \cos \theta} \]
where \( \theta = 120^\circ \) and \( \cos 120^\circ = -\frac{1}{2} \).

### Step 2: Substitute the values into the resultant formula
Substituting \( R = \sqrt{228} \) and \( \cos 120^\circ = -\frac{1}{2} \):
\[ \sqrt{228} = \sqrt{p^2 + q^2 - pq} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 228 = p^2 + q^2 - pq \quad (2) \]

### Step 4: Substitute \( q \) from equation (1) into equation (2)
From equation (1), we can express \( q \) in terms of \( p \):
\[ q = 18 - p \]
Substituting this into equation (2):
\[ 228 = p^2 + (18 - p)^2 - p(18 - p) \]

### Step 5: Expand and simplify the equation
Expanding \( (18 - p)^2 \):
\[ (18 - p)^2 = 324 - 36p + p^2 \]
Now substituting back into the equation:
\[ 228 = p^2 + (324 - 36p + p^2) - (18p - p^2) \]
Combine like terms:
\[ 228 = p^2 + 324 - 36p + p^2 - 18p + p^2 \]
This simplifies to:
\[ 228 = 3p^2 - 54p + 324 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 3p^2 - 54p + 324 - 228 = 0 \]
\[ 3p^2 - 54p + 96 = 0 \]

### Step 7: Divide the entire equation by 3
\[ p^2 - 18p + 32 = 0 \]

### Step 8: Factor the quadratic equation
Factoring gives:
\[ (p - 16)(p - 2) = 0 \]
Thus, the solutions for \( p \) are:
\[ p = 16 \quad \text{or} \quad p = 2 \]

### Step 9: Find corresponding \( q \) values
Using \( p + q = 18 \):
1. If \( p = 16 \), then \( q = 18 - 16 = 2 \).
2. If \( p = 2 \), then \( q = 18 - 2 = 16 \).

### Conclusion
The magnitudes of the forces are \( 16 \, \text{N} \) and \( 2 \, \text{N} \).