WRITE A LINEAR FUNCTION INS SLOPE INTERCEPT FORM WORD PROBLEMS

Problem 1 :

Suppose that the water level of a river is 34 feet and that it is receding at a rate of 0.5 foot per day. Write an equation for the water level L, after d days. In how many days will the water level be 26 feet?

Solution :

Let d be the number of days. 

w = 34 - 0.5 × d

Slope- intercept form :

w = -0.5d + 34

26 = -0.5d + 34

26 - 34 = -0.5d

-8 = -0.5d

d = 16

The water level will be 26 feet in 16 days.

Problem 2 :

Seth’s father is thinking of buying his son a six-month movie pass for $40. With the pass, matiness cost $1.00. If matinees are normally $3.50 each, how many times must Seth attend in order for it to benefit his father to buy the pass? 

Solution :

The cost of movies pass = $40

Matinee cost with pass = $1

Normal matinee cost = $3.50

40 + 1x = 3.50x

3.5x - x = 40

2.5x = 40

x = 40/2.5

x = 16

Therefore, Seth must attend 16 matinees in order to break even.

Problem 3 :

For babysitting, Nicole charges a flat fee of $3, plus $5 per hour. Write an equation for the cost, C, after h hours of babysitting. What do you think the slope and the y- intercept represent? How much money will she make will she make if she baby-sits 5 hours?

Solution :

Slope- Intercept form:

y = 5x + 3

Slope = 5 (cost per hour)

y- Intercept = 3 (flat fee)

If she babysits 5 hrs, she makes

y = 5(5) + 3

y = 25 + 3

y = $28

The money that should make is $28.

Problem 4 :

A plumber charges $25 for a service call plus $50 per hour of service. Write an equation in slope- intercept form for the cost, C, after h hours of service. What will be the total cost for 8 hours of work? 10 hours of work?

Solution :

Slope- Intercept form:

C = 50h + 25

The total cost for 8 hours,

C = 50(8) + 25

C = 400 + 25

C = $425

The total cost for 10 hours,

C = 50(10) + 25

C = 500 + 25

C = $525

Problem 5 :

Rufus collected 100 pounds of aluminum cans to recycle. He plans to collect an additional 25 pounds each week. Write and graph the equation for the total pounds, P, of aluminum cans after w weeks. What does the slope and y-intercept represent? How long will it take Rufus to collect 400 pounds of cans?

Solution :

The total pounds P, of aluminum cans that can be collected after w weeks is

P = 100 + 25w

The number of weeks that will be taken to collect 400 pounds of aluminum cans will be,

400 = 100 + 25w

25w = 300

w = 300/25

w = 12

Therefore, Rufus will take 12 weeks to collect 400 pounds of aluminum cans.

Problem 6 :

A canoe rental service charges a $20 transportation fee and $30 dollars an hour to rent a canoe. Write and graph an equation representing the cost, y, of renting a canoe for x hours. What is the cost of renting the canoe for 6 hours?

Solution:

Slope- Intercept form:

y = 30x + 20

The cost of renting a canoe for 6 hours,

y = 30(6) + 20

y = 180 + 20

y = 200

It will cost $200 to rent the canoe for 6 hours.

Problem 7 :

A caterer charges $120 to cater a party for 15 people and $200 for 25 people. Assume that the cost, y, is a linear function of the number of x people. Write an equation in slope-intercept form for this function. What does the slope represent? How much would a party for 40 people cost?

Solution :

Total cost = $120

Number of people = 15

120 = m × 15 + 0

120 = 15m

m = 120/15

m = 8

Cost of a party for 40 people,

y = mx + b

y = (8 × 40) + 0

y = $320

Therefore, the cost of a party for 40 people is $320.

Problem 8 :

Attorney A charges a fixed fee on $250 for an initial meeting and $150 per hour for all hours worked after that. Write an equation in slope-intercept form. Attorney B charges $150 for the initial meeting and $175 per hour. Find the charge for 26 hours of work for each attorney. Which is the better deal? At how many hours does this attorney become a better deal?

Solution :

Attorney A:

y = 150h + 250

Attorney B:

y = 175h + 150

Cost of each attorney for a 26 hour work:

Attorney A:

y = 150(26) + 250

y = $4150

Attorney B:

y = 175(26) + 150

y = $4700

Hence, the better deal is Attorney A.

Hours when Attorney A becomes the better deal:

150h + 250 < 175h + 150

250 - 150 < 175h - 150h

100 < 25h

4 < h

Therefore, Attorney A becomes the better deal at 5 hours.

Problem 9 :

A water tank already contains 55 gallons of water when Baxter begins to fill it. Water flows into the tank at a rate of 8 gallons per minute. Write a linear equation to model this situation. Find the volume of water in the tank 25 minutes after Baxter begins filling the tank.

Solution :

Slope- Intercept form:

y = 8x + 25

y = 8(25) + 25

y = 200 + 55

y = 255

The volume of water in the tank 25 minutes after Baxter begins filling the tank is 255 gallons.

Problem 10 :

A video rental store charges a $20 membership fee and $2.50 for each video rented. Write and graph a linear equation (y = mx + b) to model this situation. If 15 videos are rented, what is the revenue? If a new member paid the store $67.50 in the last 3 months, how many videos were rented?

Solution :

The total charges for x videos

y = 2.5x + 20

Graphing:

If x = 0,

y = 20

If y = 0,

2.5x = -20

x = -8

Join the points (0, 20) and (-8, 0).

If x = 15 videos,

The total revenue,

y = 20 + 2.5(15)

y = 20 + 37.5

y = $57.5

If y = 67.5,

67.5 = 20 + 2.5x

47.5 = 2.5x

x = 19