SOLUTION: i know all the parts except the last. can you please explain to me how to do that? To achieve the maximum benefit for the heart when exercising, your heart rate (in beats per minu

Question 879128: i know all the parts except the last. can you please explain to me how to do that?
To achieve the maximum benefit for the heart when exercising, your 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 your age. The upper limit is found by using 85%.
a) Find formulas for the upper and lower limits (u and I) as linear equations involving the age x.
b) what is the target heart rate zone for a 20-year-old?
c) What is the target heart rate zone for a 40-year-old?
d) Two women in an aerobics class stop to take their pulse, and are surprised to find that they have the same pulse. One women is 35 years older than the other and is working at the upper limit of her target rate zone. The younger woman is working at the lower limit of there target heart rate zone. What are the ages of the two women, and what is their pulse?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i know all the parts except the last. can you please explain to me how to do that?
To achieve the maximum benefit for the heart when exercising, your 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 your age. The upper limit is found by using 85%.
a) Find formulas for the upper and lower limits (u and I) as linear equations involving the age x.
b) what is the target heart rate zone for a 20-year-old?
c) What is the target heart rate zone for a 40-year-old?
d) Two women in an aerobics class stop to take their pulse, and are surprised to find that they have the same pulse. One women is 35 years older than the other and is working at the upper limit of her target rate zone. The younger woman is working at the lower limit of there target heart rate zone. What are the ages of the two women, and what is their pulse?
d.)

The formula for upper and lower limit is:

U = .85 * (220 - x)
L = .70 * (220 - x)
x is equal to the person's age.

Let the age of the younger person be represented by A.
Let the age of the older person be represented by B.

For the older person:

U = .85 * (220 - B).

For the younger person:

L = .70 * (220 - A).

Since the older person is working at her upper limit and the younger person is working at her lower limit and their pulse rate is the same, you get:

U of the older person equals L of the younger person.

This translate to the equation U = L which translates to:
.85 * (220 - B) = .70 * (220 - A)

Simplify both sides of this equation to get:

.85 * 220 - .85 * B = .70 * 220 - .70 * A.

Simplify further to get:

187 - .85 * B = 154 - .70 * A.

Subtract 154 from both sides of this equation and add .85 * B to both sides of this equation to get:

187 - 154 = .85 * B - .70 * A

Simplify to get:

33 = .85 * B - .70 * A

Since you know that the older person is 35 years older than the younger person, you can replace B with A + 35 in this equation to get:

33 = .85 * (A + 35) - .70 * A

Simplify this equation to get:

33 = .85 * A + .85 * 35 - .70 * A

Simplify further to get:

33 = .85 * A + 29.75 - .70 * A

Subtract 29.75 from both sides of this equation and combine like terms on the right side of this equation to get:

33 - 29.75 = .15 * A

Simplify further to get:

3.25 = .15 * A

Divide both sides of this equation by .15 to get:

3.25 / .15 = A

Simplify and commute to get:

A = 21 and 2/3

Since B = A + 35, this means that B = 56 and 2/3.

Those are the ages of the women.

The older woman's pulse rate is equal to .85 * (220 - (56 and 2/3)) = 138.833333 which is equal to 138 and 5/6.

The younger women's pulse rate is equal to .70 * (220 - (21 and 2/3)) = 138.83333 which is also equal to 138 and 5/6.

That's your answer.

The older woman is 56 and 2/3 years old.
The younger woman is 21 and 2/3 years old.
Their common pulse rate is 138 and 5/6 beats per minute.

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