Final answer:
The power (P) required for an 80.0-kg runner to maintain a constant speed of 3.85 m/s up a 9.00° incline, assuming 25% efficiency, is approximately 773 Watts.
Step-by-step explanation:
To calculate the power required by the runner, we can use the formula for power, which is given by P = F * v * cos(θ), where P represents power, F is the force exerted, v is the velocity, and θ is the angle of inclination.
The force exerted by the runner can be found using Newton's second law, F = ma, where m is the mass of the runner and a is the acceleration. As the runner is moving at a constant speed up an incline, the force required to counteract gravity's component along the incline can be calculated as F = mg * sin(θ), where g is the acceleration due to gravity (9.81 m/s²).
Given the mass of the runner as 80.0 kg, the angle of incline as 9.00°, and the velocity as 3.85 m/s, we can compute the force required. Then, by substituting the force, velocity, and the angle into the power formula, considering the 25% efficiency, we find the power exerted by the runner. This calculation results in approximately 773 Watts. This signifies the rate at which the runner must produce energy to counteract gravity and maintain a steady pace up the incline, accounting for the inefficiencies inherent in human locomotion.