MODELLING WITH LINEAR RELATIONSHIPS

Problem 1 :

In 1995, Orlando, Florida, was about 175,000. At that time, the population was growing at a rate of about 2000 per year.

Write an equation, in slope-intercept form to find Orlando’s population for any year.

i) Predict what Orlando’s population will be in 2010.

Solution :

Initial population = 175000 (y-intercept)

Growing rate = 2000 per year (slope)

y = mx + b

y = 2000x + 175000

i) The problem starts with the year 1995. At 2010, x = 15

Applying the value of x, we get

y = 2000(15) + 175000

= 30000 + 175000

= 205000

The population is 205000.

Problem 2 :

Couples are marrying later. The median age of men who tied the knot for the first time in 1970 was 23.2. In 1998, the median age was 26.7.

Write an equation, in slope intercept form to predict the median age that men marry M for any year t. 

i) Use the equation to predict the median age of men who marry for the first time in 2005.

Solution :

t is the independent variable and M is the dependent variable.

Then the linear function will be in the form, M = tx + b

(1970, 23.2) and (1998, 26.7)

Slope = (26.7 - 23.2) / (1998 - 1970)

= 3.5/28

t = 0.125

At the beginning, the mean age is 23.2. So, y-intercept is 23.2.

y = 0.125x + 23.2

M = 0.125x + 23.2

i) Median age of the men, when he marry at 2005

x = 2005 - 1970

= 35

M = 0.125(35) + 23.2

= 4.375 + 23.2

= 27.575

= 28

So, the median age is 28.

Problem 3 :

The cost for 7 dance lessons is $82. The cost for 11 lessons is $122. Write a linear equation, slope-intercept form, to find the total cost C for L lessons.

i)  Use the equation to find the cost of 4 lessons.

Solution :

Here L is the independent variable and C is the dependent variable

(7, 82) and (11, 122)

Slope = (122 - 82) / (11 - 7)

= 40/4

= 10

C = Lx + b

C = 10x + b

Applying the point (7, 82), we will get the value of b.

82 = 10(7) + b

82 = 70 + b

b = 82 - 70

b = 12

C = 10x + 12

i) Cost for 4 lessons :

When x = 4

L = 10(4) + 12

L = 40 + 12

L = 52

So, the cost of 4 lessons is $52.

Problem 4 :

It is 76º F at the 6000-foot level of a mountain, and 49º F at the 12,000-foot level of the mountain.

Write a linear equation, in slope-intercept form, to find the temperature T at an elevation e on the mountain, where e is in thousands of feet.

i) Use the equation to predict the temperature at an elevation of 20,000 feet.

Solution :

Here elevation of the mountain (e) is the independent variable and Temperature (T) is the dependent variable.

T = ex + b

(6, 76) and (12, 49)

slope = (49 - 76)/(12 - 6)

= -27/6

= -9/2

T = (-9/2) x + b

Applying the point (6, 76) to get b.

76 = (-9/2) (6) + b

76 = -27 + b

b = 76 + 27

 b = 103

T = (-9/2) x + 103

i) When the elevation is 20000, that is e = 20

T = ( -9/2) 20 + 103

T = -90 + 103

T = 13

The required temperature is 13⁰.

Problem 5 :

Between 1990 and 1999, the number of movie screens in the United States increased by about 1500 each year. In 1996, there were 29,690 movie screens.

Write an equation of a line, in slope-intercept form, to find the total number of screens y for any year x.

i) Predict the number of movie screens in the United States in 2005.

Solution :

y = mx + b

Rate of change (m) = 1500

Initial number of screens at 1996 = 29690

Applying the rate of change and y-intercept.

y = 1500x + 29690

When t = 2005 - 1996 ==> 9

Applying the value of t, we get

y = 1500(9) + 29690

= 13500 + 29690

y = 43190

At 2005, the number of screens is 43190.

Problem 6 :

A construction company charges $15 per hour for debris removal, plus a one-time fee for the use of a trash dumpster. The total fee for 9 hours of service is $195.

Write an equation of a line, in slope-intercept form, to find the total fee y for any number of hours x.

i) What is the fee for the use of a trash dumpster for 5 hours?

Solution :

Construction charge per hour (m) = 15

y = mx + b

y = 15x + b

When x = 9, y = 195

195 = 15(9) + b

b = 195 - 135

b = 60

Applying the value of b, we get

y = 15x + 60

i) When x = 5

y = 15(5) + 60

y = 75 + 60

y = 135

Fee after 5 years is $135.