A skier starts from rest and slides down a 20° incline 100 meters long. if the coefficient of friction is 0.09, how much heat is dissipated into the environment as the skier descends?

To calculate the amount of heat dissipated by the skier as they descend down the incline, we need to determine the work done against friction and then convert it into heat energy.

The work done against friction is given by the equation:

Work = Force × Distance × cos(θ)

where:
- Force is the force of friction acting on the skier
- Distance is the distance traveled by the skier down the incline
- θ is the angle of the incline

In this case, the force of friction can be calculated using the equation:

Force of friction = coefficient of friction × Normal force

where:
- The coefficient of friction is given as 0.09
- The Normal force can be determined from the gravitational force acting on the skier

The Normal force can be calculated using the equation:

Normal force = mass × gravitational acceleration × cos(θ)

In this case, the mass is not given, but we can use the known information that the skier starts from rest and slides down the incline. Assuming no energy losses due to other factors, we can use the conservation of energy to determine the mass:

Initial Potential Energy = Final Kinetic Energy

mgh = (1/2)mv^2

where:
- m is the mass (to be determined)
- g is the gravitational acceleration (approximately 9.8 m/s^2)
- h is the vertical height (sin(θ) × hypotenuse) = sin(20°) × 100 m
- v is the velocity at the bottom of the incline (to be determined)

Solving this equation for m, we get:

m = (1/2)mv^2 / (gh)

Now we know the mass, we can calculate the Normal force. Then we can substitute it into the equation for the force of friction to calculate the work done against friction.

Finally, to convert the work done into heat energy, we use the equation:

Heat energy = Work

Now let's calculate the values step by step and find the answer.