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A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after [tex]$t$[/tex] minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.

\begin{tabular}{llllll}
[tex]$t \, (\text{min})$[/tex] & 36 & 38 & 40 & 42 & 44 \\
\hline
Heartbeats & 2535 & 2675 & 2809 & 2941 & 3075
\end{tabular}

The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of [tex]$t$[/tex]. (Round your answers to one decimal place.)

The average heart rate over the interval [tex]$[36, 42]$[/tex] is [tex]$\square$[/tex]

The average heart rate over the interval [tex]$[42, 44]$[/tex] is [tex]$\square$[/tex]

An estimate for the heart rate at [tex]$t = 42$[/tex] is [tex]$\square$[/tex]


Sagot :

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."