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To calculate the average heart rate and its uncertainty, we can follow the error propagation rules, particularly focusing on the inverse relationship between time and heart rate.

### Given Data:
1. Average time between heartbeats (\(\Delta t_{\text{avg}}\)): 
   \[
   \Delta t_{\text{avg}} = 0.782 \, \text{s} \quad \pm \, 0.010 \, \text{s}
   \]

2. Formula for heart rate (\(\Delta R_{\text{avg}}\)):
   \[
   \Delta R_{\text{avg}} = \frac{60}{\Delta t_{\text{avg}}}
   \]
   
### Step 1: Calculate the average heart rate

The average heart rate is found by taking the reciprocal of the average time between heartbeats:

\[
\Delta R_{\text{avg}} = \frac{60}{0.782} \, \text{beats per minute}
\]

Let me calculate that for you.The average heart rate is approximately \( 76.7 \, \text{beats per minute} \).

### Step 2: Calculate the uncertainty in the heart rate

We need to propagate the uncertainty from \(\Delta t_{\text{avg}}\) to \(\Delta R_{\text{avg}}\). Using the error propagation formula for a function \(R = f(t)\), where \(R = \frac{60}{t}\), the uncertainty \(\Delta R_{\text{avg}}\) is given by:

\[
\Delta R_{\text{avg}} = \left| \frac{-60}{t_{\text{avg}}^2} \right| \cdot \Delta t_{\text{avg}}
\]

Now, let’s calculate the uncertainty in the heart rate.The uncertainty in the average heart rate is approximately \( \pm 0.98 \, \text{beats per minute} \).

### Final result:
\[
\Delta R_{\text{avg}} = 76.7 \, \pm 0.98 \, \text{beats per minute}
\]

Would you like further clarification or details on this process?

Here are some related questions:
1. How does the uncertainty in time affect the heart rate?
2. What is the significance of error propagation in measurements?
3. How would the result change if the average time had a larger uncertainty?
4. Can error propagation be applied to other formulas in physics?
5. What other methods can be used to reduce uncertainty in experiments?

**Tip:** Error propagation is essential for understanding the reliability of results in experiments. Always propagate uncertainties to the final result!

Use what you have learned about error propagation to determine the average heart rate (in beats per minute) and the corresponding uncertainty for your group’s data. Make sure to measure the heart rate from your measurement of Δt_avg and not from what the sensor reports directly.

Learn how to calculate the average heart rate using error propagation and determine the uncertainty based on the average time between heartbeats. This detailed analysis includes the key formula ΔR_avg = 60 / Δt_avg and covers uncertainty propagation for precise measurements, suitable for Grades 11-12.

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