I recently purchased a new Velo-ton stationary bike and took it for a spin. The bike records three key metrics throughout the ride: cadence (how fast I’m riding), resistance (how hard I have to push the pedals to keep moving) and output (the power I produce).
With a little experimentation, I determine that the power (in watts) is equal to the product of the cadence and resistance values divided by 20. For example, if my cadence is 64 and my resistance is 25, then my power output is (64·25)/20, or 80 watts.
Whenever I ride, I always make sure that my resistance is between 20 and 60, while my cadence is between 60 and 100. After a particularly grueling 30-minute workout, I notice that my average resistance was 40, while my average cadence was 80. (Note that these averages are computed per unit of time, rather than per unit of distance traveled.)
At first, I figure my average power was (40·80)/20, or 160 watts. But I soon realize other values are also possible. What is the maximum average power that I could have produced? What is the minimum?
Averageing minimum resistance and cadence with maximum resistance and cadence yields the maximum:
$$\dfrac{1}{2} \cdot \left( \dfrac{20 \cdot 60}{20} + \dfrac{60 \cdot 100}{20} \right) = 180 \text{ watts}$$Averageing minimum resistance and maximum cadence with maximum resistance and minimum cadence yields the minimum:
$$\dfrac{1}{2} \cdot \left( \dfrac{20 \cdot 100}{20} + \dfrac{60 \cdot 60}{20} \right) = 140 \text{ watts}$$