A person’s maximum heart rate is the highest rate at which their heart beats during certain extreme kinds of exercise - Leaving Cert Mathematics - Question 7 - 2010

The outlier can be observed at the coordinates (47 years, 137 bpm). Thus, the person's age is 47 years, and the maximum heart rate is 137 bpm.

Using the line of best fit shown in the diagram, the estimated maximum heart rate for a 44-year-old is approximately 176 bpm.

The slope of the line of best fit can be calculated using two points: (10, 200) and (90, 144).

The formula for the slope (m) is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting:

$m = \frac{144 - 200}{90 - 10} = \frac{-56}{80} = -0.7$

Using the point-slope form of the equation:

Let’s take one point, (10, 200):

$y - y_1 = m(x - x_1)$

Substituting values:

$y - 200 = -0.7(x - 10)$

Rearranging gives us:

$MHR = 207 - 0.7 \times (age)$

For young adults, the old rule gives a greater MHR:

• Adult aged 20: Old rule: $MHR = 220 - 20 = 200$; New rule: $MHR = 207 - 0.7 \times 20 = 193$.

For middle-aged persons, the new rule yields a greater MHR:

• Adult aged 70: Old rule: $MHR = 220 - 70 = 150$; New rule: $MHR = 207 - 0.7 \times 70 = 158$.

Using the old rule, he exercises to 75% of $MHR = (220 - 65) = 155$ bpm, giving him:

$0.75 \times 155 = 116.25 \text{ bpm} \approx 116 \text{ bpm}.$

Using the new rule, $MHR = 207 - 0.7 \times 65 hickapprox 121$ bpm, thus:

$0.75 \times 121 = 90.75 \text{ bpm} \approx 91 \text{ bpm}.$

Therefore, he should exercise to about 121 bpm for better benefits.