Let's solve the problem step-by-step, using the information in the table to find the estimated heart rate.
### 1. Average Heart Rate over the Interval [tex]\([36, 42]\)[/tex]
First, we'll calculate the average heart rate between [tex]\( t = 36 \)[/tex] minutes and [tex]\( t = 42 \)[/tex] minutes. The heartbeats at these times are 2535 and 2941 respectively.
To find the average heart rate over this interval, we use the formula:
[tex]\[
\text{Average Heart Rate} = \frac{\text{Change in heartbeats}}{\text{Change in time}}
\][/tex]
1. Calculate the change in heartbeats:
[tex]\[
\Delta \text{Heartbeats} = 2941 - 2535 = 406
\][/tex]
2. Calculate the change in time:
[tex]\[
\Delta t = 42 - 36 = 6 \text{ minutes}
\][/tex]
3. Now, calculate the average heart rate:
[tex]\[
\text{Average Heart Rate over } [36, 42] = \frac{406}{6} \approx 67.7 \text{ beats per minute}
\][/tex]
### 2. Average Heart Rate over the Interval [tex]\([42, 44]\)[/tex]
Next, we'll calculate the average heart rate between [tex]\( t = 42 \)[/tex] minutes and [tex]\( t = 44 \)[/tex] minutes. The heartbeats at these times are 2941 and 3075 respectively.
1. Calculate the change in heartbeats:
[tex]\[
\Delta \text{Heartbeats} = 3075 - 2941 = 134
\][/tex]
2. Calculate the change in time:
[tex]\[
\Delta t = 44 - 42 = 2 \text{ minutes}
\][/tex]
3. Now, calculate the average heart rate:
[tex]\[
\text{Average Heart Rate over } [42, 44] = \frac{134}{2} = 67.0 \text{ beats per minute}
\][/tex]
### 3. Estimate for the Heart Rate at [tex]\( t = 42 \)[/tex]
To estimate the heart rate at [tex]\( t = 42 \)[/tex], we'll use the average of the two average heart rates we just calculated.
1. Calculate the average of the average heart rates:
[tex]\[
\text{Heart rate at } t = 42 = \frac{67.7 + 67.0}{2} = \frac{134.7}{2} \approx 67.3 \text{ beats per minute}
\][/tex]
### Summary:
"The average heart rate over the interval [tex]\([36, 42]\)[/tex] is [tex]\(67.7\)[/tex] beats per minute."
"The average heart rate over the interval [tex]\([42, 44]\)[/tex] is [tex]\(67.0\)[/tex] beats per minute."
"An estimate for the heart rate at [tex]\( t = 42 \)[/tex] is [tex]\(67.3\)[/tex] beats per minute."
