Question

to achieve the maximum benefit for the heart when exercising, the heart rate (in beats per minute) should be in the target heart rate zone. the lower limit of this zone is found by taking 70% of the difference between 220 and the age of the person. the upper limit is found by using 85% of the difference. complete parts a through d.
a. find formulas for the upper and lower limits (u and l) as linear equations involving the age x. u = 187 - .85x (use integers or decimals for any numbers in the equation. do not factor.) l = 154 - .7x (use integers or decimals for any numbers in the equation. do not factor.)
b. what is the target heart rate zone for a 20 - year - old person? the lower limit is 140 and the upper limit is 170 beats per minute.
c. what is the target heart rate zone for a 40 - year - old person? the lower limit is 126 and the upper limit is 153 beats per minute.
d. two women in an aerobics class stop to take their pulse and find that they have the same pulse. one woman is 36 years older than the other and is working at the upper limit of her target heart rate zone. the younger woman is working at the lower limit of her target heart rate zone. what are the ages of the two women, and what is their pulse? the age of the younger woman is approximately □ years and that of older woman is approximately □ years. (round to the nearest integers as needed.)

Explanation:

Step1: Write the given equations

The upper - limit formula is $U = 187-0.85x$ and the lower - limit formula is $L = 154 - 0.7x$.

Step2: Find the target heart - rate zone for a 20 - year - old

For a 20 - year - old ($x = 20$):

• Calculate the lower limit:
• Substitute $x = 20$ into $L = 154-0.7x$.
• $L=154 - 0.7\times20=154 - 14=140$.
• Calculate the upper limit:
• Substitute $x = 20$ into $U = 187-0.85x$.
• $U=187-0.85\times20=187 - 17 = 170$.

Step3: Find the target heart - rate zone for a 40 - year - old

For a 40 - year - old ($x = 40$):

• Calculate the lower limit:
• Substitute $x = 40$ into $L = 154-0.7x$.
• $L=154-0.7\times40=154 - 28 = 126$.
• Calculate the upper limit:
• Substitute $x = 40$ into $U = 187-0.85x$.
• $U=187-0.85\times40=187 - 34 = 153$.

Step4: Solve part d

Let the age of the younger woman be $y$ and the age of the older woman be $y + 36$.

• Since their heart - rates are the same as the lower and upper limits of the target zone:
• The lower - limit equation for the younger woman is $L_y=154 - 0.7y$.
• The upper - limit equation for the older woman is $U_{y + 36}=187-0.85(y + 36)$.
• Set $154 - 0.7y=187-0.85(y + 36)$.
• Expand the right - hand side: $154 - 0.7y=187-0.85y-30.6$.
• Combine like terms: $154 - 0.7y=156.4-0.85y$.
• Add $0.85y$ to both sides: $154 - 0.7y+0.85y=156.4-0.85y+0.85y$.
• $154 + 0.15y=156.4$.
• Subtract 154 from both sides: $0.15y=156.4 - 154=2.4$.
• Solve for $y$: $y=\frac{2.4}{0.15}=16$.
• The age of the older woman is $y + 36=16+36 = 52$.

Answer:

a. The upper - limit formula is $U = 187-0.85x$ and the lower - limit formula is $L = 154 - 0.7x$.
b. For a 20 - year - old, the lower limit is 140 and the upper limit is 170.
c. For a 40 - year - old, the lower limit is 126 and the upper limit is 153.
d. The age of the younger woman is 16 years and the age of the older woman is 52 years.