To understand what \((r \cdot s)(a)\) represents, we first need to clarify the functions involved:
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Heart rate function: \(r(a) = 211 - 0.64a\) gives the heart rate \(r\) in beats per minute for a person of age \(a\) years.
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Life span function: \(s(a) = 525,600a\) provides the life span \(s\) in minutes for a person of age \(a\) years, where 525,600 is the number of minutes in a year (60 minutes per hour × 24 hours per day × 365 days per year).
Now, when we consider the product \((r \cdot s)(a)\), we are multiplying these two functions:
\[ (r \cdot s)(a) = r(a) \cdot s(a) \]
Substituting the expressions for \(r(a)\) and \(s(a)\):
\[ (r \cdot s)(a) = (211 - 0.64a) \cdot (525,600a) \]
This gives a new function that will yield a value when \(a\) (age) is input.
Interpretation of \((r \cdot s)(a)\):
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Heart Rate Times Life Span: The term \((r \cdot s)(a)\) essentially represents the total number of heartbeats a person is expected to have in their lifetime, based on their age \(a\).
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Calculation:
- The heart rate \(r(a)\) gives the number of heartbeats per minute.
- The life span \(s(a)\) gives the total duration of life in minutes.
Thus, multiplying these two quantities yields the total heartbeats over an entire lifespan, which can be calculated as:
\[ \text{Total Heartbeats} = \text{Heart Rate (beats/min)} \times \text{Life Span (min)} \]
In summary, \((r \cdot s)(a)\) gives the expected total number of heartbeats over a person's lifetime based on their age \(a\).