3.05 Interpreting linear relations
Consider the equation of a line y = mx + c.
Consider the table showing several points of a linear relationship:
Find the gradient.
Find the y-intercept.
Determine the equation of the line in gradient-intercept form.
| x | 4 | 10 | 16 |
|---|---|---|---|
| y | 12 | 24 | 36 |
Consider the following graph:
By how much does the y-value change as the x-value increases by 1?
State the gradient.
Consider the following graph:
Find the gradient.
Find the y-intercept.
Determine the equation of the line in gradient-intercept form.
Find the equations of the following lines in gradient-intercept form:
The graph shows the temperature of a room after the heater has been turned on:
Find the gradient.
Find the y-intercept.
Determine the equation of the line in gradient-intercept form.
What does the y-intercept represent?
Find the temperature of the room after the heater has been turned on for 44 minutes.
The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in its side:
Find the gradient.
Find the y-intercept.
Determine the equation of the line in gradient-intercept form.
What does the gradient represent?
What does the y-intercept represent?
Find the amount of water remaining in the bucket after 48 minutes.
A carpenter charges a callout fee of \$90 plus \$45 \text{ per hour}.
Determine the equation of the line in gradient-intercept form to represent the total amount charged, y, by the carpenter in terms of the number of hours worked, x.
Find the gradient.
What does the gradient represent?
Find the y-intercept.
What does the y-intercept represent?
Find the total amount charged by the carpenter for 3 hours of work.
Aaron is taking his new BMW out for a drive. He had only driven 70 \text{ km} in it before and is now driving it down the highway at 75 \text{ km/h}.
Determine the equation of the line in gradient-intercept form to represent the total distance, y, that Aaron had driven in his BMW in terms of the number of hours, x, from now.
Find the gradient.
What does the gradient represent?
Find the y-intercept.
What does the y-intercept represent?
Find the total distance Aaron will have driven in his BMW if his current drive begins at 6:15 pm and finishes at 8:15 pm.
A particular restaurant has a fixed weekly cost of \$2700 and receives an average of \$18 from each customer.
Determine the equation of the line in gradient-intercept form to represent the net profit, y, of the restaurant for the week in terms of the number of customers, x.
Find the gradient.
What does the gradient represent?
Find the y-intercept.
What does the y-intercept represent?
Find the restaurant's net profit if it has 300 customers for the week.
Homer is running a 90 \text{ km} ultramarathon at an average speed of 8 \text{km/h}:
Determine the equation of the line in gradient-intercept form to represent the distance Homer has left to run, y, in terms of the number of hours since the start, x.
Find the gradient.
What does the gradient represent?
Find the y-intercept.
What does the y-intercept represent?
Find the distance Homer will have left to run after 2.5 hours.
Iain has just purchased a prepaid phone, which he intends to use exclusively for sending text messages. He has also purchased some credit to use. After sending 19 text messages, he has \$38.92 of his credit remaining. After sending 26 text messages, he has \$36.68 of his credit remaining.
Determine the equation of the line in gradient-intercept form to represent the amount of credit remaining, y, in terms of the number of text messages sent, x.
Find the gradient.
What does the gradient represent?
Find the y-intercept.
What does the y-intercept represent?
Find how much credit Iain will have left after sending 43 text messages.
The following table shows the water level of a well that is being emptied at a constant rate with a pump:
| \text{Time (minutes)} | 2 | 4 | 6 |
|---|---|---|---|
| \text{Water level (metres)} | 20.2 | 16.4 | 12.6 |
Determine the equation of the line in gradient-intercept form to represent the water level, y, in terms of the minutes passed, x.
Find the gradient.
What does the gradient represent?
Find the y-intercept.
What does the y-intercept represent?
Find the water level after 12 minutes.
The following table shows the linear relationship between the length of a mobile phone call and the cost of the call:
| \text{Length of call (minutes)} | 1 | 2 | 3 |
|---|---|---|---|
| \text{Cost (dollars)} | 10.5 | 20.5 | 30.5 |
Determine the equation of the line in gradient-intercept form to represent the cost of a call, y, in terms of the length of the call, x.
Find the gradient.
What does the gradient represent?
Find the y-intercept.
What does the y-intercept represent?
Find the cost of a 11-minute call.
Consider the points in the following table. The time (x) is measured in minutes:
| \text{Time } (x\text{)} | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Temperature } (y\text{)} | 8 | 13 | 18 | 23 | 28 |
By how much does the temperature increase each minute?
What would the temperature have been at time 0?
Find the algebraic relationship between x and y.
Sketch the graph of the linear relationship.
Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol:
| \text{Number of litres} \left( x \right) | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of petrol} \left(y \right) | 0 | 14.70 | 29.40 | 44.10 | 58.80 |
Determine the equation of the line in gradient-intercept form to relate the number of litres of petrol pumped, x, and the cost of the petrol, y.
What does the gradient represent?
How much does petrol cost per litre?
How much would 14 litres of petrol cost?
After Sally starts running, her heartbeat increases at a constant rate as shown in the table below:
| \text{Number of minutes passed} \left( x \right) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate} \left( y \right) | 69 | 73 | 77 |
Complete the table.
Find the unit change in y for the above table.
Determine the equation of the line in gradient-intercept form that describes the relationship between the number of minutes passed, x, and Sally’s heartbeat, y.
What does the gradient represent?
The table below shows the linear relationship between the number of baseball bats manufactured and the total manufacturing cost:
| \text{Number of baseball bats} | 2 | 4 | 6 |
|---|---|---|---|
| \text{Cost (dollars)} | 105 | 125 | 145 |
When an extra baseball bat is produced, by how much does the cost increase?
Determine the equation of the line in gradient-intercept form to represent the total manufacturing cost, y, in terms of the number of baseball bats manufactured, x.
Find the y-intercept.
What does the y-intercept represent?
Find the total cost of manufacturing 16 baseball bats.
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 over a period of 5 minutes:
| \text{Number of minutes passed} \left( x \right) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in metres} \left( y \right) | 0 | 1.31 | 2.62 | 3.93 | 5.24 |
Find the increase in depth each minute.
Determine the equation of the line in gradient-intercept form to represent the relationship between the number of minutes passed, x, and the depth, y, of the diver.
What does the gradient represent?
At what depth would the diver be after 6 minutes?
Find the time it takes the diver to reach 9.17 \text{ m} beneath the surface.
The points show the relationship between water temperatures and surface air temperatures:
Complete the table:
| \text{Water Temperature} \left( \degree C \right) | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \text{Surface Air Temperature} \left( \degree C \right) |
Determine the linear equation relating the water temperature, x, and the surface air temperature, y.
Find the surface air temperature when the water temperature is 13 \degree \text{C}.
Find the water temperature when the surface air temperature is 37 \degree \text{C}.
It starts raining and an empty rainwater tank fills up at a constant rate of 5 \text{ litres per hour}. By midnight, there are 10 \text{ litres} of water in a rainwater tank. As it rains, the tank continues to fill up at this rate.
Complete the table:
| \text{Number of hours passed since midnight} \left( x \right) | 0 | 1 | 2 | 3 | 4 | 4.5 | 10 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank} \left( y \right) |
Determine the linear equation relating the number of hours passed since midnight, x, and the amount of water in the tank, y.
Find the y-intercept.
How many hours before midnight was the tank empty (i.e. when y = 0)? Note that x represents the number of hours passed since midnight, so a value of - x would represent x hours before midnight.
Sketch the linear equation.
There are 20 \text{ litres} of water in a rainwater tank. It rains for a period of 24 hours and during this time, the tank fills up at a rate of 8 \text{ litres per hour}.
Complete the table:
| \text{Number of hours passed } (x) | 0 | 1 | 2 | 3 | 4 | 4.5 | 10 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank }(y) |
Determine the linear equation linking the number of hours passed (x) and the amount of water in the tank (y).
Sketch the linear equation.
The number of fish in a river is approximated over a five year period. The results are shown in the following table:
| \text{Time in years} \left( t \right) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Number of fish} \left( F \right) | 6600 | 6300 | 6000 | 5700 | 5400 | 5100 |
Sketch the graph that represents this relationship.
Find the gradient.
What does the gradient represent?
Find the value of F when the line crosses the vertical axis.
Determine the linear equation.
Find the number of fish remaining in the river after 12 years.
Find the number of years until 2700 fish remain in the river.
A ball is rolled down a slope. The table below shows the velocity of the ball after a given number of seconds:
| \text{Time in seconds} \left( t \right) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Velocity in m/s} \left( V \right) | 12 | 13.8 | 15.6 | 17.4 | 19.2 | 21 |
Sketch the graph that represents this relationship.
Find the gradient.
What does the gradient represent?
What is the vertical intercept of the line?
What does the vertical intercept represent?
Determine the linear equation expressing velocity, V, in terms of time, t.
Find the velocity of the ball after 14 seconds. Express your answer as a decimal, rounded to one decimal place.
Justin is looking into the details of his mobile phone plan. He knows the costs for several call lengths as shown in the table below:
| \text{Length of call } (t \text{ minutes}) | 2 | 6 | 10 | 14 |
|---|---|---|---|---|
| \text{Cost } (C) | \$1.00 | \$2.20 | \$3.40 | \$4.60 |
Sketch the graph that represents this relationship.
Find the gradient.
Determine the linear equation that represents the cost of a call , C, in terms of the minutes passed, t.
How much will it cost to make a 8-minute call?
What is the length of a call that costs \$1.60?
A car travels at an average speed of V = 65 \text{ km/h}. The distance, D, that the car travels in time, t, is given by D=Vt.
Complete the table.
How far will it travel in 8 hours?
Find the gradient.
| t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| D |
If the destination is 455 \text{ km} ahead, how long would it take for the car to reach the destination at the given speed?
Sketch the graph that represents the relationship between D and t.
The cost of a taxi ride C is given by C = 4.5 t + 9, where t is the duration of a trip in minutes.
Find the cost of an 11-minute trip.
For every extra minute the trip takes, how much more will the trip cost?