Question

A cardiac monitor is used to measure the heart rate of a patient after surgery. it compiles the number of heartbeats after t minutes. when the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. 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 t. (give your answers correct to 1 decimal place.) t (min) 36 38 40 42 44 heartbeats 2524 2661 2800 2934 3067

Answer 1

From the given information, there are some missing tables and the correct format is outlined in the table below.

When the data in the table were graphed: the slope (tangent line) and the time in minutes are represented as follows:

t(mins)          36          38        40         42       44

heartbeats   2524    2661     2880   2934    3067

Let us assume that P refers to the point (42, 2948) which is located on the graph of heartbeats. Then, the objective is to find the slope of the secant line PQ.

where;

• Q is that point located on the graph for t = 36, 38, 40, 44

∴

For the first slope, where P = (42, 2948) & Q = (36,2524)

The slope;

$\mathbf{\Delta S = \dfrac{2524 - 2948}{36 - 42}}$

ΔS = 70.667

For the 2nd slope, where P = (42, 2948) & Q = (38,2661)

The slope;

$\mathbf{\Delta S = \dfrac{2661-2948}{38 - 42}}$

ΔS = 71.75

For the 3rd slope, where P = (42, 2948) & Q = (40,2880)

The slope;

$\mathbf{\Delta S = \dfrac{2880-2948}{40 - 42}}$  

ΔS = 34.00

For the 4th slope, where P = (42, 2948) & Q = (44, 3067)

The slope;

$\mathbf{\Delta S = \dfrac{3067-2948}{44 - 42}}$

ΔS = 59.50

Thus, the points and the slope of the secant lines can be computed as:

Points                          (36,2524)     (38,2661)        (40,2880)    (44, 3067)

Slope of secant lines    70.7              71.8                  34.0           59.5

Therefore, we can conclude that the cardiac monitor that is being used to measure the heart rate of the patient after surgery starts to decrease and reduce from 71.8 to 59.5 after being stable for a while. This shows that the heart rate is dropping after 42 minutes.

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