On the moon, the equation d = 0.8 t^{2} is used to find the distance d (in metres) that an object has fallen after t seconds. On Earth, the equation is d = 4.9 t^{2}.
A rock is dropped from a tall cliff on Earth. How far has it fallen after 5 seconds?
The same rock is dropped from a cliff on the moon. How far has it fallen after 5 seconds?
A large planet outside our solar system has gravity so great that the distance d metres fallen by an object after t seconds is given by d = 16 t^{2} and shown on the graph.
What does the point at \left(0, 0\right) represent?
After how many seconds has the object fallen approximately 200 \text{ m}? Round your answer to one decimal place.
A ball is thrown into the air at an angle. The height, y metres, of the ball at time, x seconds, is modelled by the equation \\y = 20 x - 5 x^{2} and shown on the graph.
How long does it take for the ball to return to the ground?
What does the y-value of the turning point represent?
For what x-values does this model make sense?
In a game of tennis, the ball is mistimed and hit high up into the air. Initially \left(t = 0\right), 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.
State the coordinates of the labelled points in the diagram.
A
B
C
After how many seconds will the ball come back down to a height of 2.5 \text{ m}?
A frisbee is thrown upward and away from the top of a cliff that is 240 \text{ m} tall.
The height, y metres, of the frisbee at time, x seconds, is given by the equation \\y = - 20 x^{2} + 80 x + 240.
The graph of this relationship is shown.
State the height of the frisbee at the \\y-intercept of the graph.
What does the value of the x-intercept represent?
The frisbee reaches its maximum height after 2 seconds. State the maximum height reached by the frisbee.
A rectangular enclosure is to be constructed from 100 metres of wooden fencing. The area of the enclosure is given by A = 50 x - x^{2}, where x is the length of one side of the rectangle.
Find the area of the enclosure if one side is 15 \text{ m} long.
Find the greatest possible area of the enclosure.
Hence, state the dimensions of the rectangle with this maximum area.
Ellie records the number of people standing on a train platform from midday each day. She records this information in a graph, with the vertical axis representing hundreds of people \left(P\right), and the horizontal axis representing the number of hours that have passed since midday \left(t\right).
Is the platform ever empty on the first day?
Is the platform ever empty on the second day?
The graphs of the two days were plotted together on the same axes as shown.
State whether the following are true or false:
On both days, the hour between 2 pm and 3 pm had the least number of people on the platform.
There were always more people on the platform on the second day than the first.
There were more people on the platform at midday on the second day than the first.
On both days, the hour between 4 pm and 5 pm had the least number of people on the platform.
Patricia is operating two slingshot rides at a local carnival - the Dark Horse and the Scream Machine. The trajectories of riders launched from the two slingshots are plotted on the graph below. The horizontal axis represents time t and the vertical axis represents the height h of the riders above ground.
What do the vertical intercepts of the graphs represent?
What do the horizontal intercepts of the graphs represent?
Avril says to Patricia: "I want to go on the ride that goes the highest and stays in the air the longest.". Does either ride offer this experience?
"Okay", says Avril, "I want the one that goes the highest.". Which one should Patricia recommend?
Thirty rabbits are released into the wild. In the absence of predators, the population P is expected to grow according to the function P = 24 x^{2} + 6 x + 30, where x is the number of years since release.
Calculate the rabbit population after the following number of years:
| \text{Number of years } x | 1 | 5 | 10 |
|---|---|---|---|
| \text{Population } P | ⬚ | ⬚ | ⬚ |
Consider the graph of the population over the first 6 years:
Describe the behaviour of the population growth over time.
A car that is driven along a wet road at a speed of x \text{ km/h} has a braking distance given by d = 0.008 x^{2} + 0.0045 x, where d is the distance taken to stop in metres.
Complete the table by finding the braking distance for each speed. Round your answers to the nearest whole number.
| x \text{ (km/h)} | 10 | 20 | 40 |
|---|---|---|---|
| d \text{ (m)} | ⬚ | ⬚ | ⬚ |
Using technology (or otherwise), graph the equation. How many x-intercepts does the parabola have? Consider only the domain where the model makes sense.
What does the turning point of this parabola represent?
Find the smallest x-value for which this equation makes sense.
A rectangular enclosure is to be built for an animal. Zookeepers have 26 \text{ m} of fencing, but they want to maximise 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.
The area function has been graphed. State the width that will allow the greatest possible area.
State the greatest possible area of such an enclosure.
The formula for the surface area of a sphere is S = 4 \pi r^{2}, where r is the radius in centimetres.
Complete the following table of values for S = 4 \pi r^{2}. Round your answers to two decimal places.
| r | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| S | ⬚ | ⬚ | ⬚ | ⬚ | ⬚ | ⬚ |
Sketch the graph of S = 4 \pi r^{2}.
Use the graph from part (b) to approximate the surface area of a sphere of radius 5.5 \text{ cm}.
Use the graph from part (b) to approximate the radius of a sphere that has a surface area of 804 \text{ cm}^{2}.