Find the equations of the following graphs in the given form, where c is the least positive value and x is in radians:
y = \sin \left(x - c\right) + d
y = a \cos \left(x - c\right)
y = - \sin \left(x - c\right) - d
y = - \cos \left(x + c\right) - d
The height in metres of the tide above mean sea level t hours after midnight is given by: h = 4 \sin \left(\dfrac{\pi \left(t - 2\right)}{6}\right)
Find the height above mean sea level of the tide at 1:00 am.
Find the time of the first high tide of the day.
Find the number of hours between a high tide and the following low tide.
Sketch a graph of h over the first 24 hours.
Find the predicted height of the tide at 2:00 pm.
How much higher than low tide level is the tide at 11:30 am? Round your answer to two decimal places.
Discuss the usefulness and limitations of the function.
The tide rises and falls in a periodic manner which can be modelled by a trigonometric function. Brad charts tide levels in order to determine when he can sail his ship into a bay and takes the following measurements:
- Low tide occured at 8 am, when the bay was 7 \text{ m} deep.
- High tide occured at 2 pm, when the bay was 15 \text{ m} deep.
Let t be the number of hours passed since low tide was first measured. Complete the following table of values for the tide level:
| \text{Time }(t, \text{ hours}) | 0 | 3 | 6 | 9 | 12 |
|---|---|---|---|---|---|
| \text{Tide level }(d, \text{ metres}) |
Find the amplitude of the tide level.
Find the period of the tide level.
Sketch a graph of the tide level, d, over the first 12 hours.
State which of the following equations can be used to model the tide level, d, at t hours after 8 am, where the parameters A, n and C are positive:
A \cos n t + C
- A \cos n t + C
A \sin n t + C
- A \sin n t + C
Find an equation for d.
The ship needs a depth of at least 11.7 \text{ m} to be able to sail into the bay safely. Find the number of hours Brad has after high tide before he can no longer safely sail into the bay. Round your answer to two decimal places.
The population (in thousands) of two different types of insects on an island can be modelled by the following functions:
- Butterflies: f \left( t \right) = a + b \sin \left( m t\right)
- Crickets: g \left( t \right) = c - d \sin \left( k t\right)
where t is the number of years from when the populations were first measured, and a, b, c, d, m, and k are positive constants. The graphs of f and g for the first two years are shown below:
Find the function f \left( t \right) that models the population of butterflies over t years.
Find the function g \left( t \right) that models the population of crickets over t years.
Find the number of times over an 18-year period that the population of crickets reaches its maximum.
How many years after the population of crickets first starts to increase, does it reach the same population as the butterflies?
Find t, when the population of butterflies first reaches 200\,000.
The height of water in metres on a beach wall t hours after midnight is given by:h \left( t \right) = 6 + 4 \cos \left( \dfrac{\pi}{6} t - \dfrac{\pi}{3}\right)
Write the equation in the form h \left( t \right) = 6 + 4 \cos \left( \dfrac{\pi}{6} \left(t - c\right)\right), where c is the least positive number.
Sketch a graph of h \left( t \right) over the first 24 hours.
Find the earliest time of day at which the water is at its highest.
Find the time when the water level is first 2 \text{ m} on the wall.
12 seconds into an opera song, there is a section in which the singer must alternate between high and low pitch notes for 6 seconds. During this part of the song, the volume v of the sound is given by the function: v \left( t \right) = 6 \sin \left( \dfrac{\pi}{2} t\right) + 24
Sketch a graph of the volume function for the 6-second section.
Find the maximum volume the singer’s voice reaches during the section.
What does t represent in this function?
Find the volume of the singer’s voice 2 seconds into the section.
The approximate number of hours of daylight, at a point on the Antarctic Circle is given by:L = 12 + 12 \cos \left( \dfrac{2 \pi}{365} \left(t + 10\right)\right)Where t is the number of days that have elapsed since 1 January.
Find the number of days in a year that have less than 5 hours of daylight. Round your answer to the nearest day.
The diagram shows a double-sided beacon located at point A. As it turns, the beacon lights a point on a wall that is 3 \text{ m} away. The point lit on the wall is p metres from B. If p > 0, the beacon is lighting a point to the right of B, and if p < 0 it is lighting up a point to the left of B.
If t is the time in seconds since the beacon began rotating, distance p is given by the equation:p = 3 \tan \left( \pi t\right)
Find p for each of the following, correct to two decimal places when necessary:
t = 0 seconds
t = 1.1 seconds
t = 0.7 seconds
Determine whether the following statements are true or false when t = 0.5 seconds:
At t = 0.5 seconds, p = 3 \tan \dfrac{\pi}{2} and so p is undefined.
t = 0.5 seconds is the time at which the beacon is pointing parallel to the wall, therefore no section of wall is lit.
t = 0.5 seconds is the time at which the beacon has rotated halfway around, and is therefore perpendicular to the wall, lighting up point B.
At t = 0.5 seconds, p = 3 \tan \pi and so p is equal to 0.
Three objects, X, Y and Z are placed in a magnetic field such that:
- X is 2 \text{ cm} from object Y.
- X is 4 \text{ cm} from object Z.
- As object X is moved closer to line YZ, objects Y and Z move in such a way that the lengths XY and XZ remain fixed.
Let \theta = \angle ZXY, and let the area of triangle XYZ be represented by A.
Find an equation for A in terms of \theta.
State the domain of the function of A in this context.
Sketch a graph of the function A.
Find the acute angle which will form an area that is exactly half of the maximum possible area.
A metronome is a device used to help keep the beat consistent when playing a musical instrument. It swings back and forth between its end points, just like a pendulum.
For a particular speed, the given graph represents the metronome's distance, x \text{ cm}, from the centre of its swing, t seconds after it starts swinging. Negative values of x represent swinging to the left, and positive values of x represent swinging to the right of the centre:
Find the position of the pendulum when it first started swinging.
From its starting position, did the pendulum start to swing to the left or to the right?
Find the time at which the pendulum first passes the centre of its swing.
Find the time taken to complete one full swing.
Find the function for x in terms of t.
In a car engine, pistons are connected to a crankshaft by a rod (at point R in the picture). When the engine is turned on, the piston starts to move down and up repeatedly to rotate the crankshaft.
In a particular engine, point R on the crankshaft rotates at a rate of 120 cycles each second. The distance, d, is the distance from the piston to the centre of the crankshaft in centimetres.
To model the distance d between the piston and the crankshaft at time t seconds, do we need a trigonometric, exponential or linear function?
State the period of the distance function in seconds.
The piston gets to within 1.1 \text{ cm} of the crankshaft, and as far away as 3.1 \text{ cm} from the crankshaft. Determine the amplitude of the distance function.
At time t = 0 when the engine starts, the piston is 1.1 \text{ cm} from the crankshaft. Find the equation of d.
Sketch the graph the function d = f \left( t \right).
When the piston is moving fastest, how far is it from the crankshaft?
In a car engine, pistons are connected to a crankshaft by a rod. When the engine is turned on, the piston starts to move down and up repeatedly to rotate the crankshaft.
The distance d of the piston from the centre of the crankshaft depends on the speed of the rod. In a particular engine, the piston rod rotates 60 times each second.
At time t = 0 when the engine starts, the piston is 1.4 \text{ cm} from the crankshaft.
Sketch the function d = f \left( t \right), with time t seconds on the horizontal axis.
Sounds around us create pressure waves. Our ears interpret the amplitude and frequency of these waves to make sense of the sounds.
A speaker is set to create a single tone, and the graph below shows how the pressure intensity (I) of the tone, relative to atmospheric pressure, changes over t seconds:
State whether a sine or cosine function be more suited for modelling this graph. Explain your answer.
Find the equation of the function for the intensity I after t seconds.
As a brain exercise, a music student must tap their foot every time they hear the most intense part of the sound. How many times will the student tap their foot in 14 minutes?
The change in voltage of overhead power lines over time can be modelled by a periodic function that oscillates between - 300 and 300 kiloVolts with a frequency of 40 cycles per second.
Assuming that at t = 0 seconds, the voltage is 300 \text{ kV}, state which of the following equations can be used to model voltage over time:
V = A \cos n t
V = A \sin n t
V = A \sin t
V = A \cos t
Find an equation for the voltage V as a function of time t.
Sketch a graph of the voltage over time, for 0 \leq t \leq \dfrac{1}{16}.
Describe the effect on the graph if the frequency was increased.
In Yabooma, the depth of the tide after x hours is recorded in 12-hour intervals. The depths for the first 12-hour interval are recorded in the following table:
| \text{Time } \\ \left( \text{hours} \right) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \text{Depth }\\ \left( \text{cm} \right) | 30 | 35 | 60 | 85 | 110 | 100 | 80 | 90 | 60 | 35 | 25 | 20 |
On a coordinate axes, plot the data over a 24-hour period.
Given that the points (3, 60), (6, 100), (9, 60) and (12, 20) lie on a sine curve, determine a function that models the data and is of the form a \sin \left( b \left(x - c\right)\right) + d, where a, b, c and d are constants.
Add the sketch of the graph of f \left( x \right), the function you found in part (b), to the data plotted in part (a).
In Enns, the temperature is recorded every 6 hours for 48 hours. The first nine temperatures are recorded in the following table:
On a number plane, plot the data over a two-day interval, where x is the number of hours since 12 am on Monday.
To model the data, find the equation of the sine curve that passes through the points (0, 9), (18, 16), (30, 16) and (26, 23) of the form a \sin \left( b \left(x - c\right)\right) + d, where a, b, c and d are constants.
Add the sketch of the graph of f \left( x \right), the function you found in part (b), to the data plotted in part (a).
Discuss the usefulness and limitations of the function.
| \text{Day and Time} | \text{Temperature } \left( \degree \text{C} \right) |
|---|---|
| \text{Monday }12\text{ am} | 9 |
| \text{Monday }6\text{ am} | 14 |
| \text{Monday }12\text {pm} | 22 |
| \text{Monday }6\text{ pm} | 16 |
| \text{Tuesday }12\text{ am} | 8 |
| \text{Tuesday }6\text{ am} | 16 |
| \text{Tuesday }12\text{ pm} | 23 |
| \text{Tuesday }6\text{ pm} | 17 |
| \text{Wednesday }12\text{ am} | 9 |
In Valera, the average monthly rainfall is recorded in the following table:
| \text{Month} | \text{Rainfall } \left( \text{cm} \right) | \text{Month} | \text{Rainfall } \left( \text{cm} \right) | \text{Month} | \text{Rainfall } \left( \text{cm} \right) |
|---|---|---|---|---|---|
| \text{Jan} | 1.5 | \text{May} | 9.5 | \text{Sept} | 11 |
| \text{Feb} | 1.5 | \text{June} | 11.5 | \text{Oct} | 7.5 |
| \text{Mar} | 3.5 | \text{July} | 11.5 | \text{Nov} | 4.5 |
| \text{Apr} | 7 | \text{Aug} | 12.5 | \text{Dec} | 2 |
On a coordinate axes, plot the average monthly rainfall over a two-year period, where x=1 corresponds to January of the first year.
The average rainfall can be approximated using a sine curve. Which curve best approximates the average rainfall in Valera?
For the sine curve in part (b) that best approximates the monthly average rainfall, find the following:
Amplitude
Period
Phase shift
Determine a function of the form f \left( x \right)=a \sin \left( b \left(x - c\right)\right) + d, where a, b, c and d are constants, that models the average monthly rainfall data.
Add the graph of f \left( x \right), to the coordinate axes from part (a).
Discuss the usefulness and limitations of the function.
Marge has asthma, so she is having some breathing tests done to monitor her lung capacity, V, measured in millilitres \left(\text{mL}\right). The respirologist models Marge's breathing patterns according to the model V = a \sin \left( b \left(t + c\right)\right) + d , where the quantity b \left(t + c\right) is in measured in degrees. The graph of the function is shown below:
Find the length of one full breath (inhale and exhale), in seconds.
Find the value of d, the median amount of air in Marge's lungs.
Find the value of a, the difference between the maximum and median amount of air.
Find the value of b.
If the initial volume of air in Marge's lungs was 2600 \text{ mL}, find the value of c.
Hence write the equation for function V.
Discuss the usefulness and limitations of the function.
Tobias is jumping on a trampoline. Victoria watches him bounce at a regular rate and wants to try to model his height over time. When Victoria starts her stopwatch, Tobias is at a minimum height of 30 \text{ cm} below the trampoline frame. A moment later, Victoria records Tobias reaching a maximum height of 50 \text{ cm} above the trampoline frame. To model the situation, Victoria uses the function:H \left( t \right) = a \sin \left( 2 \pi \left(t - c\right)\right) + dwhere H is the height in centimetres above the trampoline frame and t is the time in seconds and 0 < c < 1.
At what time does Tobias first reach a height of 30 \text{ cm} above the trampoline frame?
A circular Ferris wheel that is 40 \text{ m} in diameter contains several carriages. Hannah and Michael enter a carriage at the bottom of the Ferris wheel, and get off 6 minutes later after having gone around completely 3 times. When a carriage is at the bottom of the wheel, it is 1 \text{ m} above the ground.
State which of the functions below could be used to model Hannah and Michael’s height above the ground, h\left(t\right), where t is the number of minutes after getting on:
A \left(t - a\right)^{2} + k
A e^{ k t} + B
A \cos \left( k t\right) + B
A \sin \left( k t\right) + B
For the height function h\left(t\right), find:
The period.
The minimum height above ground.
The maximum height above ground.
State the equation for the height function h\left(t\right).
Sketch the graph of h\left(t\right) for the 6-minute ride.
Find the first time, t minutes, at which the passenger reaches a height of 21 \text{ m} above the ground.
A new function is formed by translating h\left(t\right) down by 21 \text{ m}. Describe what this new function represents.
A circular ferris wheel that is 102 \text{ m} in diameter contains several carriages. Valentina and Sharon enter a carriage at the bottom of the ferris wheel, and get off 24 minutes later after having gone around completely 4 times. When a carriage is at the bottom of the wheel, it is 1 \text{ m} above the ground.
Find the equation of the function that could be used to model Valentina and Sharon’s height above the ground, h\left(t\right), where t is the number of minutes after getting on.
Find the first time, t minutes, at which the passenger reaches a height of 52 \text{ m} above the ground.
Passengers on a Ferris wheel access their seats from a platform 5 \text{ m} above the ground. As each seat is filled, the Ferris wheel moves around clockwise so that the next seat can be filled. Once all seats are filled, the ride begins and lasts for 6 minutes.
The height h in metres of Amelia's seat above the ground t seconds after the ride has begun is given by:
h \left( t \right) = 14 \sin \left( 10 t - 40\right) + 16where the quantity \left( 10 t - 40\right) is in degrees.
Sketch a graph of the function h \left( t \right) for one rotation.
Find the height above the ground of Amelia's seat at the commencement of the ride. Round your answer to two decimal places.
Find the time at which Amelia first passes the access platform after the ride commences. Round your answer to two decimal places.
Find the number of times her seat passes the access platform in the first two minutes.
Due to a malfunction, the Ferris wheel stops abruptly 1 minute and 40 seconds into the ride. Find the height above the ground that Amelia is stranded at. Round your answer to two decimal places.
The height above the ground of a rider on a Ferris wheel can be modelled by the function:h \left( t \right) = 25 \cos \left( 3 \left(t - 60\right)\right) + 30where h \left( t \right) is the height in metres, t is the time in seconds, and the quantity \left( 3 \left(t - 60\right)\right) is in degrees.
Sketch a graph of h \left( t \right) for 0\degree \leq t \leq 240 \degree.
Find the maximum height of the rider.
Find the minimum height of the rider.
Find the height of the rider at t=85 seconds. Round your answer to two decimal places.
Find the time taken for the rider to complete one full revolution.