Finding the angle of the traveling direction of a sailboat

[TmrK]

Homework Statement

As a sailboat sails 55 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 42 m. What is the angle between the direction of F₁ and due north?

Homework Equations

Don't know if this pertains to adding the forces together.

The Attempt at a Solution

I assume that trigonometry is involved here to solve this problem. Other than that, that's how far I have gotten into the problem.

 

[256bits]

Well W=Fd, where the force F is in the same direction as the distance d.

You know that F1=F2 from the problem.
And W1=W2

And, from the second case the work W2=F2 x 42m

But as the sailboat is traveling due north, and a breeze is acting on the sails with a force F1 at a certain angle, west of due north. Here, the work W1 on the sailboat would be the component of F1 in the north direction and the distance 55m.

 

[TmrK]

Well W=Fd, where the force F is in the same direction as the distance d.

You know that F1=F2 from the problem.
And W1=W2

And, from the second case the work W2=F2 x 42m

But as the sailboat is traveling due north, and a breeze is acting on the sails with a force F1 at a certain angle, west of due north. Here, the work W1 on the sailboat would be the component of F1 in the north direction and the distance 55m.

So with this, W=F1(cosθ)d

And plugging in the equations,
F1(cos$\Theta$)55m = F2 X 42m (being that cos0°=1)

divide F on both sides,
cosθ55m = 42m

divide 55m with 42m, and I assume that in the end,
cosθ = .7(63 repeating)°
cosθ = .76°

Am I close?

 

[256bits]

divide 55m with 42m, and I assume that in the end,
cosθ = .7(63 repeating)°
cosθ = .76°

Am I close?

0.76° --> the little dot goes on the angle but not on the cos of the angle. Sine, cosine, tangent come from the ratios of the sides of a traingle and so dimensionless.

Close,
What angle θ will give cosθ=0.76.
They ask for the angle, and, I presume, you have to state whether or not it is east of north or west of north angle.
If you draw a diagram of the sailboat and forces you should be able to figure that out visually.

Hint;
to check if your answer is correct or not, put the values back into the equation.
ie W2 = F2 x 42

W1 = F1 cosθ x 55 = F1 0.76 x 55 = F1 x 42

Since F1 = F2, then W1 = W2 and correct.

You could do that for all problems you encounter as a check.

 

[skeptic2]

I get the impression that the author of this problem doesn't understand how sailboats work.