Problem 31 - Applied Calculus

Question

Concern the maximum heart rate (MHR), which is the maximum number of times a person's heart can safely beat in one minute. If MHR is in beats per minute and \(a\) is age in years, the formulas used to estimate MHR, are $$ \begin{array}{c} \text { For females: } \mathrm{MHR}=226-a, \\ \text { For males: } \mathrm{MHR}=220-a \end{array} $$ Recently \(^{20}\) it has been suggested that a more accurate predictor of MHR for both males and females is given by $$ \mathrm{MHR}=208-0.7 a $$ (a) At what age do the old and new formulas give the same MHR for females? For males? (b) Which of the following is true? (i) The new formula predicts a higher MHR for young people and a lower MHR for older people than the old formula. (ii) The new formula predicts a lower MHR for young people and a higher MHR for older people than the old formula. (c) When testing for heart disease, doctors ask patients to walk on a treadmill while the speed and incline are gradually increased until their heart rates reach 85 percent of the MHR. For a 65 -year-old male, what is the difference in beats per minute between the heart rate reached if the old formula is used and the heart rate reached if the new formula is used?

Step-by-Step Solution

Answer

(a) Females: 60 years, Males: 40 years. (b) Statement (ii) is true. (c) 6.375 bpm.

Step 1: Equalize formulas for females

To find the age when the old and new formulas give the same MHR for females, set the equations equal: \( 226 - a = 208 - 0.7a \). Solving this gives:\[ 226 - a = 208 - 0.7a \]Rearrange: \[ 18 = 0.3a \]\[ a = \frac{18}{0.3} = 60 \text{ years} \].

Step 2: Equalize formulas for males

To find the age when the old and new formulas are equal for males, set:\( 220 - a = 208 - 0.7a \).Solving this gives:\[ 220 - a = 208 - 0.7a \]Rearrange: \[ 12 = 0.3a \]\[ a = \frac{12}{0.3} = 40 \text{ years} \].

Step 3: Determine formula behavior for different ages

Analyze the equations to decide their behavior for young and older ages:- For young age (low \(a\)), the old formulas (226-a for females, 220-a for males) have a higher constant, potentially giving a higher MHR than the new,- For older age (high \(a\)), examine MHR at \(a = 100\): - Females: New : \(208 - 0.7 \times 100 = 138\), Old: \(126\). New is higher. - Males: New : \(138\), Old: \(120\). New is higher. Indicating the new formula predicts higher MHR for older people.

Step 4: Calculate heart rate difference for a 65-year-old male

For a 65-year-old male:- Old MHR: \(220 - 65 = 155\)- New MHR: \(208 - 0.7 \times 65 = 162.5\)- MHR at 85%: - Old: \(0.85 \times 155 = 131.75\) - New: \(0.85 \times 162.5 = 138.125\)- Difference: \(138.125 - 131.75 = 6.375\) bpm.

Key Concepts

MHR formulas

The Maximum Heart Rate (MHR) is a crucial concept for understanding cardiovascular health and physical fitness. The MHR is the maximum number of times a heart can beat in one minute. Knowing your MHR helps in determining safe and effective exercise levels.

There are several formulas to estimate MHR, which adjust according to the individual's age and gender.

• For Females: The older formula is given by: \[ \text{MHR} = 226 - a \]
• For Males: The older formula is written as: \[ \text{MHR} = 220 - a \]
• New Formula for Both Genders: An updated formula offers a more refined calculation as: \[ \text{MHR} = 208 - 0.7 \times a \]

Understanding these formulas helps in making informed decisions about your fitness routines, ensuring that the intensity of your workouts is appropriate for your age and gender. The new formula, which incorporates a constant factor multiplied by age, aims to provide a more universally applicable method, refining the estimates for both males and females.

age and heart rate

Age plays an important role in determining your maximum heart rate. As a person ages, their MHR typically decreases. This decrease in heart rate is reflected in all the formulas used to estimate MHR, as they involve subtracting a value related to age.

• In the traditional formulas, a set amount is deducted from 220 for males and 226 for females to estimate MHR.
• The newer formula introduces age as a factor multiplied by 0.7, leading to a gradual decline as age increases.

Applying these formulas to different ages, we can observe a trend:

• Younger individuals: At lower ages, the older formulas predict a slightly higher MHR as they rely on a static number with a simple subtraction.
• Older individuals: The new formula tends to predict higher MHR than the older formulas due to its less steep decline with age.

This indicates that the decline in MHR with age is more gradual in the newer formula, potentially making it more accurate for older age groups.

heart rate calculation

Calculating your MHR is straightforward once you understand the formulas. Using your age, you can simply plug it into the chosen formula to determine your MHR. Let’s see some steps for effectively performing these calculations.

• Old Method: For a 30-year-old male, the formula is 220 minus the age: \[ \text{MHR} = 220 - 30 = 190 \]
• New Method: Using the new formula for the same age: \[ \text{MHR} = 208 - 0.7 \times 30 = 187 \]

By following these calculations, individuals can better understand their heart health and adjust their exercise intensity accordingly. Always remember, the aim is to keep your heart rate within a safe and beneficial range during exercise.

heart rate in males and females

While both males and females utilize the same new formula, the older calculation methods reflect some differences between genders. The original formulas were tailored separately,

• Females: The initial estimation involved subtracting the age from 226.
• Males: Their MHR was calculated by subtracting the age from 220.

These distinctions highlight the physiological differences between the sexes. However, the new unified formula acknowledges that while differences exist, a singular formula can simplify understanding and application, potentially increasing accuracy in MHR predictions for both genders. It reflects an evolving understanding that while males and females share many similarities in cardiovascular function, the details may differ slightly. Using these formulas, anyone can adjust their exercise regime for optimal cardiovascular benefits.