Let's break down the solution step-by-step for a 41-year-old with a given exercise goal:
First, we are given the formulas to determine the lower and upper limits of the heart rate range during exercise, based on the person's age. The formulas are as follows:
- Lower limit of range: [tex]\( H = \frac{7}{10} \times (220 - a) \)[/tex]
- Upper limit of range: [tex]\( H = \frac{4}{5} \times (220 - a) \)[/tex]
Where [tex]\(a\)[/tex] is the age of the person.
1. Determine the maximum heart rate:
The maximum heart rate is calculated by subtracting the person's age from 220:
[tex]\[
220 - 41 = 179 \text{ beats per minute}
\][/tex]
2. Calculate the lower limit of the heart rate range:
Using the given formula for the lower limit:
[tex]\[
H = \frac{7}{10} \times 179
\][/tex]
Evaluating this expression:
[tex]\[
H = 0.7 \times 179 = 125.3
\][/tex]
Rounding 125.3 to the nearest integer, we get:
[tex]\[
125 \text{ beats per minute}
\][/tex]
Therefore, the lower limit of the heart rate range for a 41-year-old is approximately 125 beats per minute.
3. Calculate the upper limit of the heart rate range:
Using the given formula for the upper limit:
[tex]\[
H = \frac{4}{5} \times 179
\][/tex]
Evaluating this expression:
[tex]\[
H = 0.8 \times 179 = 143.2
\][/tex]
Rounding 143.2 to the nearest integer, we get:
[tex]\[
143 \text{ beats per minute}
\][/tex]
Therefore, the upper limit of the heart rate range for a 41-year-old is approximately 143 beats per minute.
In conclusion:
- The lower limit of the heart range for a 41-year-old is 125 beats per minute.
- The upper limit of the heart range for a 41-year-old is 143 beats per minute.
