We need to determine the magnitude of the resultant force at point C, given the following conditions:

• Force $P = 60 \, \text{lb}$ is acting vertically downward.
• A second force of $75 \, \text{lb}$ is acting at an angle of $25^\circ$ with the vertical.
• The tension in cable AC is $100.899 \, \text{lb}$.

Step 1: Break Forces into Components

First, let's break the forces into their horizontal and vertical components.

Force P:

Since $P$ acts vertically downward:

• Horizontal component: $P_x = 0$
• Vertical component: $P_y = -60 \, \text{lb}$ (negative because it is downward)

Second Force (75 lb):

Given the force is at an angle of $25^\circ$ with the vertical, we can break it into horizontal and vertical components using trigonometric functions:

• Horizontal component: $F_x = 75 \sin(25^\circ)$
• Vertical component: $F_y = 75 \cos(25^\circ)$

Step 2: Resultant Components

The total horizontal and vertical components of the resultant force are the sums of the individual components.

• Horizontal component of the resultant: $R_x = P_x + F_x = 0 + 75 \sin(25^\circ)$

• Vertical component of the resultant: $R_y = P_y + F_y = -60 + 75 \cos(25^\circ)$

Step 3: Magnitude of the Resultant

The magnitude of the resultant force $R$ can be found using the Pythagorean theorem:

$R = \sqrt{R_x^2 + R_y^2}$

Let me calculate that for you.The magnitude of the resultant force at point C is approximately $32.68 \, \text{lb}$.

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. How would the result change if the angle of the second force was 30 degrees instead of 25 degrees?
2. What effect would doubling the force $P$ have on the resultant?
3. How would you calculate the tension in the cable if the forces acting were different?
4. What role does the angle between the cable and boom play in this type of analysis?
5. How can you determine the direction of the resultant force?

Tip: Always break forces into their horizontal and vertical components to simplify resultant calculations.