Heart-Rate Recovery Model Before Driving
At a community fitness center, a member completes a 30-second sprint and then sits quietly. A coach records the member's heart rate at one-minute intervals and models it with H(t) = 72 + 100(0.82)^t, where H is heart rate in beats per minute and t is minutes after the sprint ends. The model is based on four data points from this session and is intended only for the first 15 minutes. The member's doctor has set a target: the heart rate should fall below 120 bpm before the member drives home. The coach also claims that the heart rate falls by 50 bpm within the first 2 minutes, and wants to compare the average rate of change during the first 3 minutes with the average rate of change from minute 3 to minute 6. The member's measured resting heart rate is 68 bpm, which is below the model's long-term value.
1. Solve for when the model predicts the heart rate falls below 120 bpm, and interpret the result.
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Set 72 + 100(0.82)^t < 120. Subtract 72: 100(0.82)^t < 48, so (0.82)^t < 0.48. Take logs: t ln(0.82) < ln(0.48). Since ln(0.82) is negative, divide and reverse the inequality: t > ln(0.48)/ln(0.82) = (-0.7340)/(-0.1985) = 3.70. So the heart rate falls below 120 bpm at about 3.7 minutes after the sprint. The member should wait roughly 4 minutes before driving, according to this model. Because the model is empirical, a measured reading should confirm the estimate before the member drives.
2. Evaluate the coach's claim that the heart rate falls by 50 bpm in the first 2 minutes. Compare the average rates of change on [0,3] and [3,6].
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H(0) = 72 + 100 = 172. H(2) = 72 + 100(0.82)^2 = 72 + 67.24 = 139.24. The drop is 172 - 139.24 = 32.76 bpm, not 50, so the 2-minute claim is false. H(3) = 72 + 100(0.5514) = 127.14. Average rate on [0,3] = (127.14 - 172)/3 = -14.95 bpm/min. H(6) = 72 + 100(0.3040) = 102.40. Average rate on [3,6] = (102.40 - 127.14)/3 = -8.25 bpm/min. The recovery slows because the exponential decay approaches the 72-bpm asymptote; the first interval has more room to fall, so its average rate is steeper.
3. Explain a limitation of using this model for the driving decision and for long-term heart rate.
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The model's long-term value is 72 bpm, but the member's measured resting rate is 68 bpm, so extrapolating far beyond the data would predict an unrealistic plateau above rest. The model is based on four points and is stated for the first 15 minutes; using it at t=30 or t=60 is extrapolation. An exponential model also cannot capture a subsequent rise or a floor below 72. For the driving decision, the 3.7-minute estimate is useful but should be checked with a measured reading, because individual recovery varies and the cost of being wrong is high.