To understand what \((r \cdot s)(a)\) represents, we need to analyze the functions \(r(a)\) and \(s(a)\) given in the problem:
- \(r(a) = 211 - 0.64a\) represents the heart rate of a person in beats per minute (bpm) based on their age \(a\) in years.
- \(s(a) = 525,600a\) represents the total lifespan of a person in minutes based on their age \(a\) in years, where \(525,600\) is the number of minutes in a year (60 minutes/hour × 24 hours/day × 365 days/year).
When we multiply these two functions, \((r \cdot s)(a)\), we get:
\[ (r \cdot s)(a) = r(a) \cdot s(a) = (211 - 0.64a) \cdot (525,600a) \]
This product will give us the total number of heartbeats the person has in their lifetime. Here’s how:
- \(r(a)\) gives us the heart rate in beats per minute.
- \(s(a)\) gives us the total minutes lived by that person at age \(a\).
Thus, multiplying these two gives us the total number of heartbeats completed over the entire lifespan of the person who is \(a\) years old (since it calculates the heartbeats per minute times the total minutes lived).
Therefore, \((r \cdot s)(a)\) represents:
the total number of heartbeats in a lifetime of a person who is a years old.