To calculate the change in the skier's potential energy, we need to determine the vertical distance the skier is raised and use the formula for gravitational potential energy:

1. Calculate the Vertical Height (h):
   Since the chair lift makes an angle of $14.6^\circ$ with the horizontal, we can use trigonometry to find the height: $h = d \cdot \sin(\theta)$ where:

   • $d = 2830 \, \text{m}$ (length of the lift),
   • $\theta = 14.6^\circ$.

   Substituting the values: $h = 2830 \cdot \sin(14.6^\circ) \approx 715.5 \, \text{m}$

2. Calculate the Change in Potential Energy ($\Delta PE$):
   $\Delta PE = m \cdot g \cdot h$ where:

   • $m = 75 \, \text{kg}$ (mass of the skier),
   • $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity),
   • $h \approx 715.5 \, \text{m}$.

   Substituting the values: $\Delta PE = 75 \cdot 9.8 \cdot 715.5 \approx 526087.5 \, \text{J}$

So, the change in the skier's potential energy is approximately 526,087.5 Joules.

Would you like further details on the calculations or have any questions?

1. What is potential energy and how does it relate to height?
2. How would the angle of the lift affect potential energy if it were steeper or shallower?
3. How is gravitational potential energy applied in real-world scenarios?
4. What would happen to the potential energy if the skier's mass was different?
5. How does the gravitational constant $g$ vary with altitude?

Tip: Potential energy is dependent on height, mass, and gravity, and it represents stored energy due to an object’s position in a gravitational field.