A 172 N sign is supported in a motionless position by two ropes that each make an angle of 36° with the horizontal. What force does each rope exert?

To find the force that each rope exerts, we can use trigonometry. Let's consider one of the ropes.

First, let's draw a diagram to visualize the situation. Let the sign be represented by a vertical line. Then draw two ropes attached to the sign, making an angle of 36° with the horizontal. Label the angles and forces as shown:

|
|
|----- 36°
/|
/ |
/ |
/ |
/ |
/ |
/ |
/ |
--------- <-- sign


Next, let's break down the 172 N sign weight into its horizontal and vertical components. The vertical component is equal to the weight of the sign, which is 172 N. The horizontal component is zero, as the sign is in a motionless position.

Now, let's consider the forces acting on the rope. There are two forces: the vertical component of the sign weight and the force that the rope exerts. We need to find the force that the rope exerts, so let's label it as Fr.

Next, let's consider the vertical component of the sign weight. From the diagram, we can see that this component is equal to Fr since they cancel each other out. In other words, the vertical component of the sign weight is balanced by the force that the rope exerts.

Now, we can apply trigonometry to find the force that each rope exerts. In a right triangle, the vertical component is equal to the hypotenuse multiplied by the sine of the angle. Therefore, we have:

Fr = 172 N * sin(36°)

Using a scientific calculator or a trigonometric table, we can find the sine of 36°, which is approximately 0.5878.

Fr = 172 N * 0.5878
Fr ≈ 101.06 N

Therefore, each rope exerts a force of approximately 101.06 N.