Problem 58 - Algebra and Trigonometry

Question

A force of 80 pounds on a rope is used to pull a box up a ramp inclined at \(10^{\circ}\) from the horizontal. The rope forms an angle of \(33^{\circ}\) with the horizontal. How much work is done pulling the box 25 feet along the ramp?

Step-by-Step Solution

Answer

The work done pulling the box 25 feet along the ramp is approximately \( 80 \cdot cos(23^{\circ}) \times 25 \) foot-pounds.

Step 1: Resolve the force into components

Given that the force (80 pounds) makes an angle of \(33^{\circ}\) with the horizontal, it will have a vertical and horizontal component. However, since the work being done is along the ramp (which is inclined at \(10^{\circ}\) from the horizontal), the component of the force acting in the direction of the ramp is important. This can be found using the formula: \( F_{a} = F \cdot cos(\theta) \), where \( F_{a} \) is the actual force in the direction of the ramp, \( F \) is the initial force (80 pounds in this case) and \( \theta \) is the angle between the force and the direction of motion. Note: Here \( \theta = 33^{\circ} - 10^{\circ} = 23^{\circ} \). So, \( F_{a} = 80 \cdot cos(23^{\circ}) \)

Step 2: Compute the Distance

The distance covered by the box along the ramp is directly stated in the problem as 25 feet.

Step 3: Calculate the Work Done

Work done can be computed using the formula: Work = force × distance. Here, the force is the component of the initial force in the direction of motion which was found in step 1, and the distance is the one given in the problem. So, Work = \( F_{a} \times \) distance = \( 80 \cdot cos(23^{\circ}) \times 25 \)