A rope makes an angle of 40 degrees with the ground. If the horizontal component of the tension in the rope is 100 N, what is the tension and the vertical component of the force?

The angle the rope makes with the ground is $\theta = 40^\circ$.
The horizontal component of the force is $F_x = 100 \text{ N}$.
We need to calculate the tension in the rope, $T$, and the vertical component of this force, $F_y$.

The horizontal component of the tension is given by the formula:

Substitute the given values:

To find $T$, rearrange the formula:

Using a calculator, $\cos(40^\circ) \approx 0.7660$:

The vertical component of the tension is given by the formula:

Substitute the calculated tension $T$ and the given angle $\theta$:

Using a calculator, $\sin(40^\circ) \approx 0.6428$:

The tension in the rope is $\boxed{\text{130.55 N}}$.
The vertical component of this force is $\boxed{\text{83.90 N}}$.