Justin is looking into the details of his mobile phone plan. He knows the costs for several call lengths:
| \text{Length of call } (t \text{ minutes}) | 2 | 6 | 10 | 14 |
|---|---|---|---|---|
| \text{Cost } (C) | \$1.00 | \$2.20 | \$3.40 | \$4.60 |
Plot the data on a number plane.
Is this relationship linear?
Sketch a line through the points on the number plane.
How much will it cost to make a 8-minute call?
What is the length of a call that costs \$1.60?
Is this data continuous?
Find the rule that connects the cost of a call, C, to the length of the call, t.
There are 20 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 Litres per hour.
Complete the table of values:
| \text{Number of hours passed } (x) | 0 | 1 | 2 | 3 | 4 | 4.5 | 10 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank }(y) |
Write an algebraic relationship linking the number of hours passed (x) and the amount of water in the tank (y).
Plot the points on a number plane.
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.
Robert is taking his new car for a test drive by driving straight down the highway and back. His distance y in km from the end of a 60\text{ km} highway x minutes after he takes off is given by the function y = \left| 2 x - 60\right| shown in the first graph:
In this context, what is the domain of the function?
How far does he drive in total?
How long does Robert take to reach the end of the highway?
The next day Robert goes for another drive down the same route and his distance from the end of highway is given by y = \left| 3 x - 60\right| which is shown in the second graph:
Is Robert driving faster or slower than the previous day? Explain your answer.
Suppose a musical piece calls for an orchestra to start at mezzo forte 70 decibels, then decrease in loudness to about 50 decibels in two measures, and then increase back to 70 decibels in another two measures. The sound level s can be modelled by the function \\ s = 10 | m-2| + 50 where m is the number of measures.
Graph the function for 0 \leq m \leq 7
After how many measures should the orchestra be at the loudness of "fortissimi" (90 decibels)?
While playing pool, Jane tries to shoot the eight ball into the corner pocket. Imagine that a coordinate plane is placed over the pool table as shown.
The eight ball is at (1.5, 0.5) and the pocket Jane is aiming for is at (3, 1.5). Jane is going to hit the ball off the tide of the table at (1.7, 0).
Write an equation for the path of the ball.
Will Jane make the shot?
You are trying to make a hole-in-one on the miniature golf green. Imagine that a coordinate plane is placed over the golf green as shown. The golf ball is at (3, 2) and the hole is at (9, 2). You are going to hit the ball off the top wall of the green at the point (6, 8).
Write an equation for the path of the ball.
Is it possible to make the shot? Explain your answer.
A musical group's new album is released on iTunes. Weekly sales S (in thousands) is given by S = -3|t - 10| + 50 where t is the time in weeks.
Sketch the graph of the function S.
What was the maximum number of albums sold in one week?
On Jupiter the equation d = 12.5 t^{2} can be used to approximate the distance in metres, d, that an object falls in t seconds, if air resistance is ignored.
Complete the table:
| \text{time }(t) | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| \text{distance } (d) |
Graph the function d = 12.5 t^{2} on a number plane.
Use the equation to determine the number of seconds, t, that it would take an object to fall 84.4\text{ m}. Round the value of t to the nearest second.
A rectangular enclosure is to be built for an animal. Zookeepers have 26 metres of fencing, but they want to maximize the area of the enclosure. Let x be the width of the enclosure.
Form an expression for the length of the enclosure in terms of x.
Form an expression for A, the area of the enclosure, in terms of x.
The graph of the area function is shown:
What width will make the greatest possible area?
What is the greatest possible area of such an enclosure?
In a game of cricket, the ball is hit high up into the air. Initially the ball is struck 2.5\text{ m} above the ground and hits the ground 6 seconds later. It reaches its greatest height 2 seconds after being hit. The graph shows the path of the ball where t is the number of seconds after the ball is hit and y is the height of the ball above the ground:
Find the coordinates of:
A
B
C
The path has an equation of the form y = a \left(t - h\right)^{2} + k. Find the value of h.
Use point A to form an algebraic relationship between a and k.
Use point C to form another algebraic relationship between a and k.
Find the value of a.
Find the value of k, the greatest height reached by the ball.
Large sprinklers are used to water crops. When water is sprayed, it takes the path of a parabola. The stream of water reaches a greatest height of 16.2 metres above the ground, 18 metres from the sprinkler.
Let the position of the sprinkler be \left(0, 0\right). Let x be the horizontal distance from the sprinkler and y be the height of the water above the ground.
The projection of the water from the sprinkler is shown in the graph. Find the equation of the parabola.
If the sprinkler were to rotate 360 \degree about the point (0, 0), what area of crops would it irrigate?
The arch of a parabolic bridge starts 18 \text{ m} to the left of its centre and ends 18 \text{ m} to the right of its centre, and the peak of its arch is 108 \text{ m} above the road:
Determine the coordinates of the following points:
Point A
Point B
Find an equation for y, the height of the arch above the road, at distance x metres from the centre of the bridge.
Pedestrians can climb along the arch up to a vertical point that is half the height of the arch. At their highest point, exactly how far horizontally are pedestrians from the centre of the arch?
The mass in grams, M, of a cube of cork is given by the formula M = 0.28 x^{3}, where x is the side length of the cube in centimetres.
Find the mass of a cubic centimetre of cork? Round your answer to two decimal places.
Complete the following table. round your answers to two decimal places.
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| M |
Sketch the graph of M.
A cube has side length 7 \text{ cm} and a mass of 1715 \text{ g} . The mass of the cube is directly proportional to the cube of its side length.
Let k be the constant of proportionality for the relationship between the side length x and the mass m of a cube. Find the value of k.
Hence state the equation relating the mass, m, and side length, x, of a cube.
Sketch the graph of the functions for m.
From the equation, find the mass of a cube with side 8.5 \text{ cm}, to the nearest gram.
A cube has a mass of 1920 \text{ g}. From the graph, find the whole number value its side length is closest to.
The volume of a sphere has the formula V = \dfrac{4}{3} \pi r^{3}. The graph relating r and V is shown:
A sphere has radius measuring 4 \text{ m}.
Using the graph, state the interval that the volume lies within.
Using the graph, what is the radius of a sphere of volume 108 \pi \text{ m}^{3}?
As elevation (A metres) increases, atmospheric air pressure (P pascals) decreases according to the equation A = 15200 \left(5 - \log P\right).
Trekkers are attempting to reach the 8850 m elevation of Mt Everest’s summit. When they set up camp at night, their barometer shows a reading of 45611 pascals. How many more vertical metres do they need to ascend to reach the summit?
Round your answer to the nearest metre.
A communications company found that the more they spend on advertising, the higher their revenue. Their sales revenue, in thousands of dollars, is given by \\ R = 10 + 20 \log_{4} \left(x + 1\right), where x represents the amount they spend on advertising (in thousands of dollars).
Determine their sales revenue if they spend no money on advertising.
Determine their sales revenue if they spend \$14\,000 on advertising. Round your answer to the nearest thousand.
Would you say that every extra \$1000 spent on advertising becomes more or less effective in terms of raising revenue? Explain your answer.
Researchers conducted a test to determine how well information is retained through the method of rote learning. They asked students to memorise mathematical formulae in the lead up to the first test, and then study no further. They continued to test them once a month over 7 months. They found that the average student’s test scores (P) decreased over time (t months), but at a slowing rate as shown in the following graph:
Let t be the number of months that have passed since the first test, and P be the average student's test score. Which equation could be used to model their findings of the relationship between P and t under the rote learning method?
P = - 86 - 23 \log_{2} \left(t + 1\right)
P = 23 \log_{2} \left(t + 1\right) - 86
P = 86 - 23 \log_{2} \left(t + 1\right)
P = 23 \log_{2} \left(t + 1\right) + 86
Using this equation for P, what was the average student’s test score on the initial test at t = 0?
What was the average student’s test score on the 6th test? Round your answer to the nearest whole number.
Find the value of how many months it takes for the average student’s test score to equal 0 using the rote learning method. Write your answer as a whole number of months.
The number of registered nurses working in hospitals t years after the year 2002 can be modelled by the equation N = 28 \log_{4} \left(t + 2\right), where N represents the number of nurses in thousands.
According to the model, how many registered nurses were working in hospitals in the year 2002?
What was the number of registered nurses in 2004?
The signal ratio D measured in decibels of an electronic system is given by the formula D = 10 \log\left(\dfrac{F}{I}\right) where F and I are the output and input powers measured in megawatts, MW, of the system respectively.
Find the input power I in MW if the output power is equal to 10 MW and the signal ratio is 20 decibels.
The graph on the right shows the equation D = 10 \log \left(\dfrac{F}{5}\right) when the input power is 5 MW.
What interval of D values contains the signal ratio when F = 9?
What interval of F values contains the output power when the signal ratio is D = 1?
The magnitude of an earthquake is measured by values on the Richter scale. A 1 unit increase on the Richter scale represents a 10-fold increase in the strength of the earthquake, so that a 4.0 earthquake is 10 times larger than a 3.0 earthquake. The function relating Richter Scale Measure x and Microns of Ground Motion y is given by y = 10^{x}.
Complete the table of values:
| \text{Richter Scale Measure } (x) | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| \text{Microns of Ground Motion } (y) |
Graph the function on a number plane.
According to the model, can the Microns of Ground Motion ever be 0?
Two earthquakes are measured on the Richter scale, one measuring 1 and another measuring 3. How many times larger was the earthquake measuring 3?
Chile experienced an earthquake followed by an aftershock. The earthquake itself measured 4 and the aftershock measured 3.8 on the Richter scale. How many times stronger was the initial earthquake? Round your answer to one decimal place.
Consider the graph shown:
State whether the following relationships could be represented by the given graph.
The number of people, y, attending a parent/teacher conference when there are x parents, each bringing 2 children.
The number of layers, y, resulting from a rectangular piece of paper being folded in half x times.
The number of handshakes, y, made by x people in a room if every person shakes hands with every other person.
Find the equation of the graph.
Use this equation to determine the total resulting thickness if a rectangular piece of paper of thickness 0.02\text{ mm} is folded 11 times.
If the electricity bill is not paid by the due date, the company charges a fee for each day that it is overdue. The following table shows the fees:
| \text{Number of days after bill due } (x) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \text{Overdue fee in dollars } (y) | 4 | 8 | 16 | 32 | 64 | 128 |
At what rate is the fee increasing each day?
If the bill is paid 3 days overdue, what overdue fee will it incur?
Find the equation that models the overdue fee, y, as a function of the number of days overdue, x.
A \$128 bill is overdue. How many days must it remain overdue for the overdue penalty to equal the cost of the bill itself?
The mass in kilograms, M, of a baby orangutan at n months of age is given by the equation M = 1.8 \times 1.1^{n}, for ages up to n = 6 months.
What is the mass, M, of a baby orangutan at 3 months of age to one decimal place.
Complete the table. Round your answers to one decimal place.
| \text{months } (n) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| \text{mass } (M) |
Graph the function on a number plane.
The population, P, of a town in millions after t years is approximated by the formula P = 8 \left(1 + \dfrac{2}{100}\right)^{t}
Complete the table of values, round your answers correct to one decimal place:
| t | 5 | 10 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|
| P |
Sketch the graph of the function P = 8 \left(1 + \dfrac{2}{100}\right)^{t}.
After how many years will the population reach 12.6 million?
Consider the exponential functions P given by y = 9 \left(3^{ - x }\right) and Q given by y = 5 \left(3^{ - x }\right).
Are the exponential functions P and Q increasing or decreasing?
Sketch the two functions on the same set of axes.
As x tends to \infty, what value does each function approach?
Describe a transformation of the graph y=3^{-x} that would obtain:
The equation A = 12\,000 \times 1.15^{n} can be used to calculate the value, A, of a \$12\,000 investment at 15\% p.a., with interest compounded annually, after n years.
Calculate the value of the investment after 2 years.
Complete the table below:
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| A |
Sketch the graph of the function.
How many years, n, does it take for the balance to reach \$20\,988.08?
A car originally valued at \$28\,000 is depreciated at the rate of 15\% per year. The salvage value, S, of the car after n years is given by S = 28\,000 \left(1 - \dfrac{15}{100}\right)^{n}.
Complete the following table. Round your answers to two decimal places.
| n | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|
| S |
Sketch the graph of S = 28000 \left(1 - \dfrac{15}{100}\right)^{n}
What is the value of the car after 3 years?
After how many years does the value of the car drop down to an amount of \$6\,485.27?
To investigate the environmental effect on bacterial growth, two colonies of the same bacteria were studies. One was placed in a constantly sunlit environment, while the other in a dark environment. The graph shows the population, P, of each colony after a certain number of days, d.
How many more bacteria were initially present in the sunlit environment?
By what percentage rate did the bacteria in sunlight increase each day?
By what percentage rate did the bacteria in darkness increase each day?
Compare the growth rates of the two groups of bacteria.
The population of two different bacteria, labelled J and K, are given by the tables below:
Bacteria J:
| t\text{ (time in days)} | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P \text{ (population)} | 1 | 60 | 3\,600 | 2.16 \times 10^{5} | 1.296 \times 10^{7} |
Bacteria K:
| t\text{ (time in days)} | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Q \text{ (population)} | 1 | 50 | 2\,500 | 1.25 \times 10^{5} | 6.25 \times 10^{6} |
Does the population of bacteria J increase slower or faster than the population of K?
The population P of bacteria J at time t can be modelled using the general equation P \left( t \right) = a^{t}, where a > 1. By using the table of values, graph P \left( t \right) for t > 0.
The population Q of bacteria K at time t can be modelled using the general equation Q \left( t \right) = b^{t}, where b > 1. By using the table of values, graph Q \left( t \right) for t > 0.
Can the population of bacteria K be found by dilating the population of bacteria J by a factor?
A group of people are trying to decide whether to charter a yacht for a day trip to the Great Barrier Reef. The total cost of chartering a yacht is \$1200. The cost per person if n people embark on the trip is C = \dfrac{1200}{n}.
Complete the following table of values:
| n | 1 | 2 | 4 | 6 | 8 | 10 | 12 | 14 |
|---|---|---|---|---|---|---|---|---|
| C \, (\$) |
Sketch the graph of C = \dfrac{1200}{n}
Alternatively, the day tour costs \$120 per person to run. Using the graph or otherwise, determine how many people will be needed to charter the yacht so that the two options cost the same for each person.
A group of architecture students are given the task of designing the layout of a house with a rectangular floorplan. There are no restrictions on the length and the width of the house, but the floor area must be 120 square meters. Each student will be allocated a rectangle with a different pair of dimensions to any other student.
Complete the table for the various widths given:
| \text{Width in meters } (x) | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| \text{Length in meters } (y) |
Form an equation for y in terms of x.
As the width of the house increases, what happens to the length of the house?
If the width is 24 \text{ m}, what will be the length of the floor area?
Sketch the graph of the relationship between x and y.
An experiment was carried out on a given mass of gas enclosed at constant temperature. The results are given in the table:
Plot the results on a number plane.
What pressure would be required to halve the volume to 12\text{ cm}^3
Find the equation relating P and V.
| \text{Pressure}, P \text{ (cm Hg)} | \text{Volume}, V (\text{cm}^{3}) |
|---|---|
| 200 | 24 |
| 150 | 32 |
| 100 | 48 |
| 80 | 60 |
| 50 | 96 |
Suppose you attend a fundraiser where each person is given a ball, each with a different number. The balls are numbered 1 through x. Each person places his or her ball in an urn. After dinner, a ball is chosen at random from the urn. The probability that your ball is selected is \dfrac{1}{x}. Therefore, the probability that your ball is not chosen is 1 - \dfrac{1}{x}.
Sketch the graph of P \left( x \right) = 1 - \dfrac{1}{x}.
As the number of people in attendance, x, increases, does the likelihood of your ball not being chosen increase or decrease?