The maximum and minimum resultant of two forces acting at a point are 10N and 6N respectively. If each force is increased by 3N, find the resultant of new forces when acting at a paint at an angle of `90^(@)` with each-other :-

Options

A. `sqrt(146)N`
B. 11N
C. `sqrt(70)N`
D. 8N

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We know that the maximum resultant \( R_{max} \) of two forces \( F_1 \) and \( F_2 \) is 10 N, and the minimum resultant \( R_{min} \) is 6 N.

### Step 2: Set Up the Equations
From the properties of forces, we can establish the following equations:
1. \( R_{max} = F_1 + F_2 \)
2. \( R_{min} = |F_1 - F_2| \)

Substituting the given values:
1. \( F_1 + F_2 = 10 \) (Equation 1)
2. \( |F_1 - F_2| = 6 \) (Equation 2)

### Step 3: Solve the Equations
From Equation 1:
\[ F_1 + F_2 = 10 \]

From Equation 2, we can consider two cases:
- Case 1: \( F_1 - F_2 = 6 \)
- Case 2: \( F_2 - F_1 = 6 \)

**Case 1:**
1. \( F_1 - F_2 = 6 \) (Equation 3)

Now, we can solve Equations 1 and 3 together:
- From Equation 1: \( F_1 + F_2 = 10 \)
- From Equation 3: \( F_1 - F_2 = 6 \)

Adding these two equations:
\[ (F_1 + F_2) + (F_1 - F_2) = 10 + 6 \]
\[ 2F_1 = 16 \]
\[ F_1 = 8 \, \text{N} \]

Substituting \( F_1 \) back into Equation 1:
\[ 8 + F_2 = 10 \]
\[ F_2 = 2 \, \text{N} \]

**Case 2:**
1. \( F_2 - F_1 = 6 \) (Equation 4)

Now, we can solve Equations 1 and 4 together:
- From Equation 1: \( F_1 + F_2 = 10 \)
- From Equation 4: \( F_2 - F_1 = 6 \)

Adding these two equations:
\[ (F_1 + F_2) + (F_2 - F_1) = 10 + 6 \]
\[ 2F_2 = 16 \]
\[ F_2 = 8 \, \text{N} \]

Substituting \( F_2 \) back into Equation 1:
\[ F_1 + 8 = 10 \]
\[ F_1 = 2 \, \text{N} \]

### Step 4: Increase Each Force by 3 N
Now we have:
- \( F_1 = 8 \, \text{N} \) and \( F_2 = 2 \, \text{N} \) or vice versa.

If we increase each force by 3 N:
- New \( F_1 = 8 + 3 = 11 \, \text{N} \)
- New \( F_2 = 2 + 3 = 5 \, \text{N} \)

### Step 5: Calculate the Resultant of the New Forces
Since the forces are acting at an angle of \( 90^\circ \) with each other, we can use the Pythagorean theorem to find the resultant \( R \):
\[ R = \sqrt{F_1^2 + F_2^2} \]
\[ R = \sqrt{11^2 + 5^2} \]
\[ R = \sqrt{121 + 25} \]
\[ R = \sqrt{146} \]
\[ R \approx 12.08 \, \text{N} \]

### Final Answer
The resultant of the new forces when acting at a point at an angle of \( 90^\circ \) with each other is approximately \( 12.08 \, \text{N} \).

---