2.06 Modelling linear equations
Some friends spent a total of \$5.00 on stickers and badges at a store. A sheet of stickers costs \$0.50 and a badge costs \$1.50.
Write a linear equation that represents the relationship between the number of sheets of stickers, x, and the number of badges, y, the friends bought for \$5.00.
If the friends bought 2 badges, how many sheets of stickers did they buy?
The cost of a taxi ride is given by C = 5.5 t + 3, where t is the duration of the trip in minutes.
Calculate the cost of an 11-minute trip.
For every extra minute the trip takes, how much more will the trip cost?
What could the constant value of 3 represent in context?
A baseball is thrown vertically upward by a baseball player when he is standing on the ground, and the velocity of the baseball, V (in yards per second), after T seconds is given by V = 120 - 32 T.
Complete the table of values:
| \text{Time} | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Vertical velocity} |
State the slope of the linear function.
Explain the negative value of V when T = 4.
The amount of medication M (in milligrams) in a patient’s body gradually decreases over time t (in hours) according to the equation M = 1050 - 15 t.
After 61 hours, how many milligrams of medication are left in the body?
Calculate the number of hours it will take for the medication to be completely removed from the body.
A carpenter charges a callout fee of \$150 plus \$45 per hour.
Write a linear equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.
Find the total amount charged by the carpenter for 6 hours of work.
Mohamad is taking his new Subaru out for a drive. He had only driven 50 miles in it before and is now driving it down the highway at 75\text{ mi/h} .
Write a linear equation to represent the total distance, y, that Mohamad had driven in his Subaru as a function of the number of hours, x.
Find the total distance Mohamad will have driven in his Subaru if his current drive begins at 5:10 pm and finishes at 7:25 pm.
The table shows the linear relationship between the length of a cellphone call and the cost of the call:
| \text{Length of call in minutes} \left(x\right) | 1 | 2 | 3 |
|---|---|---|---|
| \text{Cost in dollars } (y) | 7.6 | 14.4 | 21.2 |
Write a linear equation to represent the cost of a call, y, as a function of the length of the call, x.
Find the cost of a 6-minute call.
Mario is running a 100 \text{ km} ultramarathon at an average speed of 9 \text{ km/h}.
Write a linear equation to represent the distance Mario has left to run, y, as a function of the number of hours since the start, x.
Find the distance Mario will have left to run after 4.5 hours.
A particular restaurant has a fixed weekly cost of \$1300 and receives an average of \$16 from each customer.
Write an equation to represent the net profit, y, of the restaurant for the week as a function of the number of customers, x.
Find the restaurant's net profit if it has 310 customers for the week.
A cellphone salesman earned \$600 in a particular week during which he sold 26 phones and \$540 in another week during which he sold 20 phones.
Write a linear equation to represent the weekly earnings of the salesman, y, as a function of the number of phones sold, x.
Find how much the salesman will earn in a week during which he sells 36 phones.
In a study, scientists found that the more someone sleeps, the quicker their reaction time. The table below displays the findings:
| \text{Number of hours of sleep } (x) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Reaction time in seconds } (y) | 6 | 5.8 | 5.6 | 5.4 | 5.2 | 5 |
How much does the reaction time decrease for each extra hour of sleep?
Write a linear equation relating the number of hours of sleep, x, and the reaction time, y.
Calculate the reaction time for someone who has slept 4.5 hours.
Calculate the number of hours someone sleeps if they have a reaction time of 5.5 seconds.
A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table below shows the depth of the diver, in yards, over 4 minutes:
| \text{Number of minutes passed }\left(x\right) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in yards }\left(y\right) | 0 | 1.4 | 2.8 | 4.2 | 5.6 |
Calculate the increase in depth each minute.
Write a linear equation for the relationship between the number of minutes passed, x, and the depth, y, of the diver.
Calculate the depth of the diver after 6 minutes.
Calculate how long the diver takes to reach 12.6 \text{ yd} beneath the surface.
After Mae starts running, her heartbeat increases at a constant rate.
Complete the following table:
| \text{Number of minutes passed } (x) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate } (y) | 49 | 55 | 61 | 67 | 73 | 79 |
What is the unit change in y for the above table?
Write a linear equation that describes the relationship between the number of minutes passed, x, and Mae’s heartbeat, y.
Describe what the slope represents in this context.
A racing car starts the race with 150 \text{ L} of fuel. From there, it uses fuel at a rate of 5\text{ L} per minute.
Complete the following table of values:
| \text{Number of minutes passed } (x) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| \text{Amount of fuel left in the tank } (y) |
Write a linear equation linking the number of minutes passed, x, and the amount of fuel left in the tank, y.
How many minutes will it take for the car to run out of fuel?
Paul has just purchased a prepaid phone, which he intends to use exclusively for sending text messages, and has purchased some credit along with it to use. After sending 11 text messages, he has \$34.39 of credit remaining and after sending 19 text messages, he has \$30.31 of credit remaining.
The relationship between the number of text messages sent and the amount of credit remaining is linear. Determine the slope of the linear function.
Write a linear equation to represent the amount of credit remaining, y, as a function of the number of text messages sent, x.
Describe what the slope of the line represents in this context.
State the value of the y-intercept.
Describe what the y-intercept represents in this context.
Find how much credit Paul will have left after sending 36 text messages.
The table shows the water level of a well that is being emptied at a constant rate with a pump:
| \text{Time in minutes } \left(x\right) | 2 | 5 | 8 |
|---|---|---|---|
| \text{Water level in meters } \left(y\right) | 26.8 | 25 | 23.2 |
Write a linear equation to represent the water level, y, as a function of the minutes passed, x.
State the slope of the function.
Describe what the slope of the function represents in this context.
State the y-intercept.
Describe what the y-intercept represents in this context.
Calculate the water level be after 15 minutes.
Kerry currently pays \$50 a month for her internet service. She is planning to switch to a fibre optic cable service.
Write an equation for the total cost T of Kerry's current internet service over a period of n months.
For the fibre optic cable service, Kerry pays a one-off amount of \$1200 for the installation costs and then a monthly fee of \$25. Write an equation of the total cost T of Kerry's new internet service over n months.
Complete the table of values for the total cost of the current internet service.
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T\text{ (dollars)} |
Complete the table of values for the total cost of the fibre optic cable service.
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T\text{ (dollars)} |
Hence, determine how many months it will take for Kerry to break-even on her new internet service.