Question

The heart rate of a person can be modeled by the function r(a)=211−0.64a where a is the person's age in years and r(a) is heartbeats per minute. The life span in minutes of a person is modeled by the function s(a)=525,600a . Explain what (r⋅s)(a) represents

Answer 1

To understand what \((r \cdot s)(a)\) represents, we first need to clarify the functions involved:

1. **Heart rate function**: \(r(a) = 211 - 0.64a\) gives the heart rate \(r\) in beats per minute for a person of age \(a\) years.

2. **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 \((r \cdot s)(a)\)**:

1. **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 \(a\).

2. **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 \(a\).