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3 LINEAR MODELS
An arithmetic sequence is a sequence with a recurrence relation given by
an = an− 1 + d,
where d is a constant (called the common difference). Assuming the starting index of this arithmetic sequence is 0, we have
a 1 = a 0 + d a 2 = a 1 + d = a 0 + d + d a 3 = a 2 + d = a 0 + d + d + d … an = a 0 + d + d + … + d = dn + a 0
A model based on an arithmetic sequence is called a discrete linear model since the isolated points of the graph of an = dn + a 0 lie on a line with slope d and y-intercept a 0. A continuous linear model is one with a continuous independent variable and has equation y = mx + b.
EXAMPLE 1. A certain long-distance telephone company charges $2. for a collect call lasting up to one minute and $0.65 for each additional minute or fraction thereof.
a. Let n denote the length of the call in minutes, and let pn denote the price of the collect call in dollars. Find a linear model.
b. How much does a collect call lasting 6 minutes and 12 seconds cost?
c. How long can a collect call last if you do not want its cost to exceed ten dollars?
EXAMPLE 2. Over a 15-year period, a piece of equipment depreciates linearly from its original cost of $20,000 to its scrap value of $2000.
a. Find a linear model. Be sure to declare all variables.
b. By how much does the equipment depreciate each year?
c. How much is the equipment worth 11 years after being bought?
c. Assuming that the same linear trend applies to an "average" male whose height is more than 6 feet, what weight should the "average" male who is 6 feet 5 inches tall have?
EXAMPLE 4. The following table gives some measurements for the rate of chirping (in chirps per minute) of the striped ground cricket at various Fahrenheit temperatures.
Temp (°F) 89 72 93 84 81 75 70 82
Chirps 78 60 79 73 63 62 59 68
Temp (°F) 69 83 80 83 81 84 76
Chirps 61 65 60 69 64 68 57
a. Create a scatter plot.
Store the temperatures in L 1. Store the number of chirps in L 2. In the Stat Plot menu, select Plot1 and set up a scatter plot of the data.
In the Zoom menu, select ZoomStat.#
(^) First make sure that all functions are cleared in the Y= menu.
b. Use linear regression to find a linear model expressing the number of chirps per minute (c) in terms of the Fahrenheit temperature (F). Round values to 2 places after the decimal point. Also recreate the scatter plot with the regression line included.
In the Stat menu, CALC submenu, enter the command LinReg(ax+b) L 1 ,L 2 ,Y 1 .## The regression equation is stored under Y 1 in the Y= menu.
Press the GRAPH key.
c. If you hear a striped ground cricket make 85 chirps per minute, what temperature do you think it is? Round to the nearest degree.
(^) Y 1 can be found in the VARS menu, Y-VARS submenu, Function sub-submenu.
- The following table gives data on the IQ and GPA of 10 students at a prestigious university.
IQ 100 120 110 105 85 95 130 100 105 90 GPA 3.0 3.8 3.1 2.9 2.6 2.9 3.6 2.8 3.1 2.
a. Using linear regression, find an equation expression GPA (y) in terms of IQ (x). Round values to 4 places after the decimal point.
b. What is the best estimate of the GPA of a student with IQ 115? Round to 2 places after the decimal point.
c. What is the best estimate of the IQ of a student having GPA 2.40? Round to the nearest integer.
ANSWERS
1a. dn = dn− 1 + 60, d 0 = 125
1b. dn = 60n + 125
1c. $
2a. cn = −2.5n + 334.
2b. 162 cans of soda per day
2c. 91 cents per can
3a. $3892.
3b. t = 0.27n − 3654
3c. $11,
3d. $50,
4a. y = 0.0285x + 0.
4b. 3.
4c. 82
