To calculate the energy expended by Jack's bike, we need to consider the work done against gravity and the work done against friction.
First, let's calculate the work done against gravity. The gravitational force is given by the formula F_gravity = mass × acceleration due to gravity, where the mass is the sum of the bike's mass and Jack's mass. Assuming a gravitational acceleration of approximately 9.8 m/s^2, the gravitational force can be calculated as follows:
F_gravity = (213 kg + Jack's mass) × 9.8 m/s^2
Next, we need to calculate the displacement in the vertical direction. Given that Jack rides the bike up a 30.0 degree slope for a distance of 25.3 m, the vertical displacement can be calculated as follows:
Vertical displacement = 25.3 m × sin(30.0°)
Now, let's calculate the work done against gravity:
Work against gravity = F_gravity × Vertical displacement
The next thing to consider is the work done against friction. The question states that the frictional force opposing the bike's movement is one-tenth its weight. We can calculate the frictional force as follows:
Frictional force = (1/10) × F_gravity
Finally, let's calculate the total work done by adding the work against gravity and the work against friction:
Total work done = Work against gravity + (Frictional force × 25.3 m)
Now, you have all the information to calculate the energy the bike expends. Note that the mass of 213 kg mentioned in the question might need to be adjusted depending on whether it includes Jack's mass or not.