To calculate the average power output of the person as they run up the slope, we need to determine the work done and the time taken.

Here's the approach we'll take:

1. Calculate the Change in Potential Energy (PE): Since the person moves up the slope, there's an increase in gravitational potential energy.

   • Mass, $m = 73 \, \text{kg}$
   • Distance along the slope, $d = 23 \, \text{m}$
   • Angle of slope, $\theta = 16^\circ$
   • Gravitational acceleration, $g = 9.81 \, \text{m/s}^2$

   The vertical height $h$ gained is: $h = d \sin \theta = 23 \cdot \sin(16^\circ)$

   Now calculate the change in potential energy: $\Delta \text{PE} = m \cdot g \cdot h$

2. Calculate the Change in Kinetic Energy (KE): The person starts from rest and reaches a final velocity of $v = 3.0 \, \text{m/s}$. $\Delta \text{KE} = \frac{1}{2} m v^2$

3. Total Work Done: The work done $W$ is the sum of the changes in potential and kinetic energy: $W = \Delta \text{PE} + \Delta \text{KE}$

4. Calculate the Average Power Output: Power $P$ is the work done over time $t = 7.4 \, \text{s}$: $P = \frac{W}{t}$

Solution

Let's go through the calculations step-by-step.

1. Calculate $h$: $h = 23 \cdot \sin(16^\circ) \approx 6.34 \, \text{m}$

2. Calculate $\Delta \text{PE}$: [ \Delta \text{PE} = 73 \cdot 9.81 \cdot 6.34 \approx