Problem 56 - Algebra and Trigonometry

Question

A wagon is pulled along level ground by exerting a force of 25 pounds on a handle that makes an angle of \(38^{\circ}\) with the horizontal. How much work is done pulling the wagon 100 feet? Round to the nearest foot-pound.

Step-by-Step Solution

Answer

The work done pulling the wagon 100 feet, when rounded to the nearest foot-pound, is approximately \(\lfloor 25 \cdot \cos(38^\circ) \cdot 100 \rfloor\) foot-pounds.

Step 1: Analyze the known values

First identify all the given values. We know that the force exerted is 25 pounds, the angle made with the horizontal is \(38^\circ\), and the distance covered is 100 feet.

Step 2: Calculate the horizontal component of the force

In order to calculate work done, we must consider only the force component that acts in the direction of motion. Here, the direction of motion of the wagon is along the horizontal axis. Therefore, we need to determine the horizontal component (Fh) of the force applied. We use the formula \(Fh = F \cdot \cos(\theta)\), where F is the force and \(\theta\) is the angle. Substituting the given values into the formula we get \(Fh = 25 \cdot \cos(38^\circ)\).

Step 3: Calculate the work done

We calculate work done using the formula \(Work = Force \cdot Distance\). However, remember that here, the force we consider is the horizontal component of the force we calculated in the previous step. So, we replace \(Force\) with \(Fh\) in the formula, which gives us \(Work = Fh \cdot Distance\). Substituting the values for \(Fh\) and Distance, we get \(Work = 25 \cdot \cos(38^\circ) \cdot 100\) foot-pounds.

Step 4: Round the result

Finally, since the problem asks for the answer to the nearest foot-pound, we round the calculated work done to the nearest whole number.