Two lifeguards pull on ropes attached to a raft. If they pull in the same direction, the raft experiences a net external force of 363 N to the right. If they pull in opposite directions, the raft experiences a net external force of 147 N to the left.

Draw a free body diagram for each situation and find the magnitude of the larger of the two individual forces. Answer in units of N.

a + b = 363

a - b = 147
----------- add
2 a = 510

a = 255 N

In order to solve this problem step-by-step, let's first draw the free body diagram for each situation.

1. When the lifeguards pull in the same direction:
In this case, we have two forces acting on the raft - the force applied by the first lifeguard (F1) and the force applied by the second lifeguard (F2). Since the net external force is to the right, the forces applied by the lifeguards should also be pointing to the right.

F1 ------------>
Raft
F2 ------------>

2. When the lifeguards pull in opposite directions:
In this case, we again have two forces acting on the raft - the force applied by the first lifeguard (F1) and the force applied by the second lifeguard (F2). However, since the net external force is to the left, the forces applied by the lifeguards should be pointing in opposite directions.

F1 <------------
Raft
F2 ------------>

Now, let's find the magnitude of the larger of the two individual forces.

1. When the lifeguards pull in the same direction:
Since the net external force is 363 N to the right, we know that the magnitude of the larger force should be 363 N.

2. When the lifeguards pull in opposite directions:
Since the net external force is 147 N to the left, we know that the magnitude of the larger force should be 147 N.

So, the magnitude of the larger of the two individual forces is 363 N when the lifeguards pull in the same direction, and 147 N when they pull in opposite directions.

To find the magnitude of the larger of the two individual forces, we can set up a system of equations using the given information.

Let's assume that each lifeguard exerts a force of F₁ and F₂ in the same and opposite directions, respectively.

When they pull in the same direction, the net external force is 363 N to the right. This means the equation would be:
F₁ - F₂ = 363 N

When they pull in opposite directions, the net external force is 147 N to the left. This gives us the equation:
F₁ + F₂ = -147 N

Now, we can solve this system of equations to find the values of F₁ and F₂.

Adding the two equations together, we get:
2F₁ = 216 N

Dividing both sides by 2, we find:
F₁ = 108 N

Substituting this value of F₁ into either of the initial equations, we can solve for F₂:
108 N + F₂ = -147 N

Subtracting 108 N from both sides, we get:
F₂ = -255 N

Since we are looking for the magnitude of the larger force, we take the absolute value of F₂:
|F₂| = |-255 N| = 255 N

Therefore, the magnitude of the larger of the two individual forces is 255 N.