A 89.7-kg skier rides a 3040-m-long lift to the top of a mountain. The lift makes an angle of 14.3 ° with the horizontal. What is the change in the skier's gravitational potential energy?

To determine the change in the skier's gravitational potential energy, we need to calculate the difference in height between the top and bottom of the lift.

First, let's find the vertical height the skier has ascended. We can use trigonometry to find the vertical component of the lift's displacement.

The vertical component of the lift's displacement can be calculated using the formula:

Vertical displacement = Displacement × sin(θ)

where:
Displacement = 3040 m (length of the lift)
θ = 14.3 ° (angle of the lift with the horizontal)

Now, let's substitute the given values into the formula:

Vertical displacement = 3040 m × sin(14.3 °)

To calculate this, we need to convert the angle to radians. Since sin function takes radians as input, we'll convert 14.3 ° to radians:

14.3 ° × (π / 180 °) = 0.249 rad

Now, substituting the values:

Vertical displacement ≈ 3040 m × sin(0.249 rad) ≈ 795.3 m

The vertical displacement is approximately 795.3 m.

Next, we need to calculate the change in gravitational potential energy. The change in gravitational potential energy can be calculated using the formula:

Change in potential energy = m × g × h

where:
m = 89.7 kg (mass of the skier)
g = 9.8 m/s^2 (acceleration due to gravity)
h = vertical displacement (795.3 m)

Now, substituting the values into the formula:

Change in potential energy = 89.7 kg × 9.8 m/s^2 × 795.3 m

Calculating this, we get:

Change in potential energy ≈ 696,220 J

Therefore, the change in the skier's gravitational potential energy is approximately 696,220 Joules.