Consider the graph which shows the cost of a consultation with a medical specialist for a student or an adult, according to the length of the consultation:
Find the cost for an adult consultation of 9 minutes.
Find the cost for a student consultation of 9 minutes.
Determine the percentage discount for a student consultation.
David decides to start his own yoga class. The cost and revenue functions of running the class have been graphed:
Calculate the amount of revenue David receives for each student.
Determine the number of students that must attend the class so that David can cover his costs.
Calculate the profit David makes if there are 8 students in his class.
The conversion rate between the Australian dollar and the Euro is 1 AUD =0.7 EUR.
Graph the relationship between the Australian Dollar and the Euro, with AUD on the horizontal axis and EUR on the vertical axis of a number plane.
Calculate the gradient of the line.
Using the graph, convert 5 AUD into EUR.
Using the graph, convert 10.5 EUR into AUD.
Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body.
Consider the graph and table below, which show the amount of medication in the bloodstream for the liquid and capsule form of the medication:
Liquid form
Capsule form
| \text{Time (mins)}, t | \text{Amount in} \\ \text{blood (mgs)}, A |
|---|---|
| 4 | 24.6 |
| 7 | 42.3 |
| 10 | 60 |
| 13 | 77.7 |
Use the graph to find the rate, in \text{mg} per minute, in which the liquid form is absorbed.
Use the table to find the rate, in \text{mg} per minute, in which the capsule form is absorbed.
In which form is the medication absorbed more rapidly?
The travel graph of John and Kate is shown below:
Calculate the difference in the amount of time travelled for John and Kate.
Calculate John's average speed in \text{km/h} for the trip. Round your answer to two decimal places.
Calculate Kate's average speed in \text{km/h} for the trip.
At what time had Kate and John travelled the same distance?
At the time when Kate and John had travelled the same distance, what was the actual distance travelled?
The graph shows a straight line that approximates the global life expectancy of a child over a period of one hundred years. The graph shows the life expectancy to be 53 years in 1910, and 69 years in 2010.
Estimate the life expectancy of a child in 1960.
A clothing manufacturer is deciding whether to employ people or to purchase machinery to manufacture their line of t-shirts. After conducting some research, they discover two equations that model the two options, where y is the cost and x is the number of t-shirts to be made:
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Cost of using people: y = 800 + 60 x
-
Cost of using machinery: y = 3200 + 20 x
Use the graphing function of your CAS calculator to graph the two cost functions in the same number plane. Use this graph to find x, the number of t-shirts to be produced, at which the cost of using people or machines is the same.
State the range of values of x for which it will be more cost efficient to use machines to manufacture the t-shirts.
State the range of values of x for which it will be more cost efficient to employ people to manufacture the t-shirts.
Justin is looking into the details of his mobile phone plan. He knows the costs for several call lengths as shown in the table:
| \text{Length of call, }t\text{ mins} | 2 | 6 | 10 | 14 |
| \text{Cost, }C | \$1.00 | \$2.20 | \$3.40 | \$4.60 |
Plot the given data on a number plane and connect the points with a line.
How much will it cost to make an 8-minute call?
What is the length of a call that costs \$1.60?
Determine the rule that connects the cost of a call, C, to the length of the call, t.
Charlie is on a holiday and has been spending money at a rate of \$50 per day. After 13 days of the holiday, he has \$450 remaining as shown in the following table:
| \text{Day, }d | 4 | 9 | 10 | 13 |
|---|---|---|---|---|
| \text{Money remaining, }t | \$900 | \$650 | \$600 | \$450 |
Plot the given data on a number plane and connect the points with a line.
How much money did Charlie have at the beginning of the trip?
How much money did Charlie have after the 18th day?
At the end of which day did Charlie have \$850 remaining?
At the end of which day did Charlie have \$100 remaining?
Determine the rule that connects the amount of money remaining, t, to the days away on holiday, d.
To measure the effectiveness of a new train timetable, the average waiting time after a train was due at Southern Cross Station was recorded for 4 months in the following table:
| \text{Month} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Average waiting time (seconds)} | 30 | 50 | 70 | 90 |
Let x be the number of months and y be the average waiting time in seconds.
Plot the given data on a number plane and connect the points with a line.
What will the average waiting time be in 6 months?
What will the average waiting time be in 1 year?
After how many months will the average waiting time be 170 seconds?
After how many months will the average waiting time be 230 seconds?
Determine the rule that connects the average waiting time in seconds, y, to the number of months, x, that have passed.
A racing car starts the race with 250 litres of fuel. From there, it uses fuel at a rate of 5 litres per minute.
Complete the table of values:
| \text{Number of minutes passed, }x | 0 | 5 | 10 | 15 | 20 | 50 |
|---|---|---|---|---|---|---|
| \text{Amount of fuel left in tank, }y |
Determine an algebraic rule linking the number of minutes passed, x, and the amount of fuel left in the tank, y.
Describe how the amount of fuel in the car is changing over time.
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, }t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Velocity, }V | 12 | 13.3 | 14.6 | 15.9 | 17.2 | 18.5 |
Determine the rule that connects the velocity, V, to the time in seconds, t.
Use your CAS calculator to graph the line that represents the relationship between velocity and time.
Describe the meaning of the gradient of the line in this context.
Describe the meaning of the vertical intercept of the line in this context.
Find the velocity of the ball after 19 seconds, rounded to one decimal place.
Kerry currently pays \$50 a month for her internet service. She is planning to switch to a fibre optic cable service.
Complete the table of values for the total cost of the current internet service:
Write an equation for the total cost, T, of Kerry's current internet service over a period of n months.
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T \, (\$) |
For the fibre optic cable service, Kerry must pay a one-off amount of \$1200 for the installation costs and then a monthly fee of \$25.
Complete the table of values for the total cost of the fibre optic cable service:
Write an equation for the total cost T of Kerry's new internet service over n months.
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T \,(\$) |
Sketch the pair of lines that represent the costs of the two internet services on a number plane.
Determine how many months it will take for Kerry to break even on her new internet service.
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 5 minutes:
| \text{Number of minutes passed, }x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in metres, }y | 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 metres beneath the surface.
After Mae starts running, her heart rate in beats per minute increases at a constant rate as shown in 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 |
Determine Mae's heart rate after 12 minutes.
Calculate the change in heart rate per minute.
Write an equation that describes the relationship between the number of minutes passed, x, and Mae’s heart rate, y.
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.
State the gradient of the linear function.
State the meaning of the gradient in this context.
State the value of the y-intercept.
State the meaning of the y-intercept in this context.
Find the total amount charged by the carpenter for 6 hours of work.
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 gradient of the linear function.
Write an equation to represent the amount of credit remaining, y, as a function of the number of text messages sent, x.
State the meaning of the gradient in this context.
State the value of the y-intercept.
State the meaning of the y-intercept in this context.
Find how much credit Paul will have left after sending 36 text messages.
A phone salesperson earned \$1200 in a particular week during which she sold 22 units and \$1350 in another week during which she sold 27 units.
Let x be the number of units sold and y be the weekly earnings.
Write the linear equation that models the earnings of the salesperson in terms of the units sold.
Use your equation to predict the earnings of the salesperson if she sells 35 units.
Luke purchases 30\text{ L} of mixed leaded and unleaded petrol solution, which costs him a total of \$103.20. The price of leaded petrol is \$3.54\text{/L}, while the price of unleaded petrol is \$2.94\text{/L}.
Let x and y be the number of litres of leaded and unleaded petrol that make up the solution respectively.
Use the fact that Luke purchased a total of 30\text{ L} of petrol to write an equation involving x and y.
Use the fact that Luke purchased a total of \$103.20 worth of petrol to set up a second equation involving x and y.
Use the graphing function of your CAS calculator to sketch the graphs of the two equations on the same number plane.
Hence determine the amount of leaded petrol that was used in the solution.
Determine the amount of unleaded petrol that was used in the solution.
A circle with centre C\left(11, 13\right) has a diameter with end points A\left(5, 14\right) and B\left(p, q\right).
Find the value of p.
Find the value of q.
Find the equation of the line passing through B with gradient \dfrac{9}{2}.
A\left( - 2 , - 1 \right), B\left(0, 0\right), and C\left(1, k\right) are the vertices of a right-angled triangle with right angle at point B.
Find the value of k.
Find the area of the triangle.
Consider the triangle shown:
Find the gradient of side AB.
Find the gradient of side AC
Find the exact length of the side AB
Find the exact length of the side AC
Hence, name the type of triangle that has been graphed.
ABCD is a rhombus as shown on the number plane:
Find the gradients of the two diagonals, AC and BD.
Calculate the product of the two gradients.
Hence, what property of a rhombus has been proven?
The vertices of \triangle ABC are A\left(9, - 12 \right), B\left(4, 4\right) and C\left( - 8 , - 5 \right). The sides AB and AC have midpoints D and E respectively.
Find the coordinates of points D and E.
Find the gradient of side BC.
Find the gradient of side DE.
Are BC and DE parallel?
Consider the quadrilateral on the number plane:
At first glance, what type of quadrilateral could ABCD be?
Determine the gradient of:
AD
AB
BC
DC
Calculate the exact length of sides:
Hence, classify this quadrilateral.
The four points A\left(2, - 7 \right), B\left(5, - 3 \right), C\left(10, - 5 \right) and D\left(7, - 9 \right) are the vertices of a quadrilateral.
Plot the quadrilateral on a number plane.
Find the gradient of:
AB
BC
CD
DA
State the parallel sides.
Classify the quadrilateral ABCD.