Questions: As a sailboat sails 54 m due north, a breeze exerts a constant force F1 on the boat's sails. This force is directed at an angle west of due north. A force F2 of the same magnitude directed due north would do the same amount of work on the sailboat over a distance of just 47 m. What is the angle between the direction of F1 and due north?

As a sailboat sails 54 m due north, a breeze exerts a constant force F1 on the boat's sails. This force is directed at an angle west of due north. A force F2 of the same magnitude directed due north would do the same amount of work on the sailboat over a distance of just 47 m. What is the angle between the direction of F1 and due north?
Transcript text: As a sailboat sails 54 m due north, a breeze exerts a constant force $\vec{F}_{1}$ on the boat's sails, This force is directed at an angle west of due north. A force $\overrightarrow{F_{2}}$ of the same magnitude directed due north would do the same amount of work on the sailboat over a distance of just 47 m. What is the angle between the direction of $\vec{\Gamma}_{1}$ and due north?
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Solution

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Solution Steps

Step 1: Analyze the problem

The problem states that the force $\vec{F_1}$ does the same amount of work as the force $\vec{F_2}$ over a shorter distance. The magnitudes of the forces are the same. The work done by a force is given by $W = Fd\cos\theta$, where $F$ is the magnitude of the force, $d$ is the distance, and $\theta$ is the angle between the force and the displacement.

Step 2: Set up equations for work

Let the magnitude of both forces be $F$. The displacement for $\vec{F_1}$ is 54 m due north. The angle between $\vec{F_1}$ and the displacement is $\theta$. The work done by $\vec{F_1}$ is $W_1 = F(54)\cos\theta$. The displacement for $\vec{F_2}$ is 47 m due north. The angle between $\vec{F_2}$ and the displacement is 0 degrees since the force is due north. The work done by $\vec{F_2}$ is $W_2 = F(47)\cos(0) = 47F$.

Step 3: Solve for the angle

The problem states $W_1 = W_2$. Therefore, $54F\cos\theta = 47F$. Dividing both sides by $F$, we have $54\cos\theta = 47$. Then, $\cos\theta = \frac{47}{54}$. Finally, $\theta = \arccos(\frac{47}{54})$. Calculating this gives $\theta \approx 28.7^\circ$.

Final Answer The final answer is $\boxed{28.7}$

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