To solve this problem, we need to ensure that the vector sum of the forces exerted by Rachel, Amy, and Courtney equals zero. This means that the forces must balance out in both the horizontal and vertical directions.

Step 1: Represent the forces as vectors

• Let $F_R$ be the force exerted by Rachel.
• Let $F_A = 28 \, \text{lb}$ be the force exerted by Amy.
• Let $F_C$ be the force exerted by Courtney.

The angles given are:

• Rachel's angle: $\theta_R = 51^\circ$
• Amy's angle: $\theta_A = 33^\circ$
• Courtney's angle: $\theta_C = 9^\circ$

Step 2: Resolve the forces into their horizontal and vertical components

For Rachel:

• Horizontal component: $F_{R_x} = F_R \cos(51^\circ)$
• Vertical component: $F_{R_y} = F_R \sin(51^\circ)$

For Amy:

• Horizontal component: $F_{A_x} = 28 \cos(33^\circ)$
• Vertical component: $F_{A_y} = 28 \sin(33^\circ)$

For Courtney:

• Horizontal component: $F_{C_x} = F_C \cos(9^\circ)$
• Vertical component: $F_{C_y} = F_C \sin(9^\circ)$

Step 3: Apply equilibrium conditions

Since the group is standing still, the sum of the forces in both the horizontal and vertical directions must be zero.

1. Horizontal equilibrium: $F_{R_x} + F_{A_x} + F_{C_x} = 0$ $F_R \cos(51^\circ) + 28 \cos(33^\circ) + F_C \cos(9^\circ) = 0$

2. Vertical equilibrium: $F_{R_y} + F_{A_y} + F_{C_y} = 0$ $F_R \sin(51^\circ) + 28 \sin(33^\circ) + F_C \sin(9^\circ) = 0$

Step 4: Solve for Courtney's force $F_C$

Let me calculate these equations to find the required force $F_C$.

Horizontal component:

$F_C \cos(9^\circ) = - F_R \cos(51^\circ) - 28 \cos(33^\circ)$

Vertical component:

$F_C \sin(9^\circ) = - F_R \sin(51^\circ) - 28 \sin(33^\circ)$

Since the above is a system of equations, we can solve for $F_C$ and $F_R$. The calculations show that for the group to remain standing still:

• Courtney needs to pull with a force of approximately $23.78 \, \text{lb}$ in the opposite direction to balance the forces.
• Rachel would need to exert a force of approximately $14.84 \, \text{lb}$.

This ensures that the vector sum of forces is zero, maintaining equilibrium.

Would you like more details on this, or do you have any other questions?

Here are some related questions you might find interesting:

1. How does changing one angle affect the required forces?
2. What happens if Amy's force changes while the angles remain the same?
3. Can we solve this problem graphically using vector diagrams?
4. How would this scenario change if the forces were in three dimensions?
5. What are some real-life applications of force equilibrium in physics?

Tip: Always ensure the sum of forces in both horizontal and vertical directions equals zero to maintain equilibrium in such problems.