A person stands at rest on a slope which makes an angle 16 degrees to the horizontal. They have a mass of 73 kg. They start to run up the slope. After a time of 7.4 s they reach a distance 23 m up the slope and have a velocity of 3.0 m/s.

(c) Calculate the average power output of the person.

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 

A person stands at rest on a slope which makes an angle 16 degrees to the horizontal. They have a mass of 73 kg. They start to run up the slope. After a time of 7.4 s they reach a distance 23 m up the slope and have a velocity of 3.0 m/s. (c) Calculate the average power output of the person.

Learn how to calculate the average power output of a person with a mass of 73 kg running up a 16° slope for 7.4 seconds, covering a distance of 23 m. This physics problem includes concepts such as potential and kinetic energy and uses trigonometry to determine the change in height.

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