WRITE A LINEAR MODEL FROM A WORD PROBLEM

Write an equation, in slope-intercept form, to model each situation.

Problem 1 :

You rent a bicycle for $20 plus $2 per hour.

Solution :

y = mx + b

Rate of change (m) = $2 per hour

y-intercept = $20

y = 2x + 20

Problem 2 :

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

a) Predict what Orlando’s population will be in 2010

Solution :

Population at 1995 = 175000

Growth rate (m) = 2000 per year

y = mx + b

y = 2000x + 175000 -----(1)

a) Population at 2010 :

Each year the increase in population is 2000.

Difference between 1995 to 2010 is 15 years.

By applying x = 15 in (1), we get

y = 2000(15) + 175000

y = 30000 + 175000

y = 205, 000

Problem 3 :

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.

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

Solution :

Let x the year and y be the age,

(1970, 23.2) and (1998, 26.7)

Rate of change or Slope (m) = (26.7 - 23.2)/(1998 - 1970)

Rate of change (m) = 3.5/28

m = 0.125

y = 0.125x + b

y-intercept (b) = 23.2

y = 0.125x + 23.2

Difference between 1970 to 2005 is 35

y = 0.125(35) + 23.2

y = 27.57

So, the median age is 27.57 at 2005.

Problem 4 :

Carmela is a member of a social club. She pays an annual membership fee and $15 for each event she attends. The equation y = 15x + 25 represents her total cost each year. Which statement(s) about the function is true?

(Hint: three of these are true)

A. The initial value is 15.

B. x represents the cost of each event.

C. The rate of change is 15.

D. The initial value represents the annual membership fee.

E. The number of events she attends is a function of the total cost.

F. The total cost is a function of the number of events she attends.

Solution :

By comparing the equation y = 15x + 25 with slope intercept form, we get

rate of change (m) = 15 and y-intercept (b) = 25

A. The initial value is 15.              False

B. x represents the cost of each event.     True

C. The rate of change is 15.            True

D. The initial value represents the annual membership fee.

True

E. The number of events she attends is a function of the total cost.        False

F. The total cost is a function of the number of events she attends.                  False

Problem 5 :

Vinh pays a convenience fee when he reserves movie ticket son his cell phone app. The app shows him the total cost of his purchase for different number of tickets in the table shown.

a) What is the equation that models this linear function? Show your work.

b) How much is each movie ticket?

Solution :

Let x be the number of tickets and y be the total cost.

y = mx + b ----(1)

(2, 32) and (3, 44.50)

Slope (or) rate of change (m) = (44.5 - 32) / (3 - 2)

m = 12.5

By applying m = 12.5 in (1), we get

y = 12.5x + b

Applying the point (6, 82), we get

82 = 12.5(6) + b

82 = 75 + b

b = 82 - 75

b = 7

So, the required equation is y = 12.5x + 7.

b) Cost of 1 ticket :

x = 1

y = 12.5(1) + 7

y = 12.5 + 7

y = 19.5

Problem 6 :

You and your friend are riding bicycles down the same road at different constant speeds. After 3 minutes, you are 60 feet ahead of your friend. After 5 minutes, you are 100 feet ahead of your friend.

a. Write an equation that represents the distance y (in feet) you are ahead of your friend after x minutes.

b. How far ahead will you be after 10 minutes?

Solution :

a)

Choosing two points on the line (3, 60) and (5, 100)

Slope = (100 - 60) / (5 - 3)

= 40 / 2

= 20

Equation of the line :

y = mx + b

Applying m = 20 and the point (5, 100), we get

100 = 20(5) + b

100 = 100 + b

b = 0

So, the required equation representing the situation is y = 20x

b) Distacne covered after 10 minutes 

y = 20(10)

y = 200 feet

Problem 7 :

You are knitting a blanket at a rate of 4 inches per day. After 4 days, you have knitted 16 inches.

a. Write an equation that represents the number of inches y you have knitted after x days.

b. How many inches will you have knitted after 7 days?

c. Will you have knitted 100 inches after 30 days?

Solution :

Slope = 4 inches per day

y = mx + b

Applying the point (4, 16), we get

16 = 4(4) + b

16 = 16 + b

b = 16 - 16

b = 0

a) the required equation is y = 4x

b) After 7 days

y = 4(7)

y = 28 inches

c) after 30 days

y = 4(30)

= 120 inches

So, after 30 days blanket can be knited for more than 100 inches. Then it is yes.