Target Heart Rate (Refer to the preceding exercise.)
A target heart rate T that is half a person’s maximum
heart rate is given by T=110-1/2A, where A
is a person’s age.
(a) What is T for a person 30 years old? 50 years old?
(b) Sketch a graph of the system of inequalities.
T>110-1/2A
T<220-A
Assume that A is between 20 and 60.
(c) Interpret your graph.
(a) For a person 30 years old, we can substitute A = 30 into the formula T = 110 - (1/2)A:
T = 110 - (1/2)(30)
T = 110 - 15
T = 95
So, for a person 30 years old, T = 95.
For a person 50 years old, we can substitute A = 50 into the same formula:
T = 110 - (1/2)(50)
T = 110 - 25
T = 85
So, for a person 50 years old, T = 85.
(b) Here's a sketch of the graph showing the system of inequalities T > 110 - (1/2)A and T < 220 - A, with A ranging from 20 to 60:
^
T > 220 - A|
| \
| \
| \
| \ T = 220 - A
T > 110 - (1/2)A|
| \
| \
| \
| \
|_________\______> A
20 60
(c) The graph illustrates the range of target heart rates T for individuals between the ages of 20 and 60. The line T = 220 - A represents the upper limit, while the line T = 110 - (1/2)A represents the lower limit. The shaded area above the line T = 220 - A and below the line T = 110 - (1/2)A represents the valid range of target heart rates. As age (A) increases, the target heart rate (T) decreases.
To find the target heart rate for a person of a certain age, you can use the formula T = 110 - (1/2)A, where A represents the person's age.
(a) To find the target heart rate for a person who is 30 years old, substitute A = 30 into the formula:
T = 110 - (1/2)(30)
T = 110 - 15
T = 95
Therefore, the target heart rate for a person who is 30 years old is 95.
To find the target heart rate for a person who is 50 years old, substitute A = 50 into the formula:
T = 110 - (1/2)(50)
T = 110 - 25
T = 85
Therefore, the target heart rate for a person who is 50 years old is 85.
(b) To sketch a graph of the system of inequalities T > 110 - (1/2)A and T < 220 - A, you can use a coordinate plane with the horizontal axis representing the age (A) and the vertical axis representing the target heart rate (T).
On the graph, mark the points (30, 95) and (50, 85) as two separate points. Then draw a line connecting these two points to represent the target heart rates between the ages of 30 and 50.
Additionally, draw a dashed line at T = 110 - (1/2)A and another dashed line at T = 220 - A. These dashed lines represent the boundaries of the inequalities.
(c) The graph illustrates the target heart rates for individuals between the ages of 30 and 50. The line connecting the points represents the target heart rates that fall between the two lines T = 110 - (1/2)A and T = 220 - A.
Any point on the line represents an age and the corresponding target heart rate that satisfies the inequalities T > 110 - (1/2)A and T < 220 - A. Any point below the line represents an age and target heart rate that does not satisfy the inequalities.