The length of a spring, cm , at time seconds, is modelled by the function
Write down
Give one criticism of this model for large values of .
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The length of a spring, cm , at time seconds, is modelled by the function
Write down
Give one criticism of this model for large values of .
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A dolphin is swimming such that it is diving in and out of the sea at a constant speed.
The height, cm, of the dolphin, relative to sea level , at time sconds, is to be modelled using the formula where are constants.
On each jump and dive the dolphin reaches a height of 70 cm above sea level and a depth of 70 cm below sea level.
Write down the value of .
Starting at sea level, the dolphin takes seconds to jump out of the water, dive back under and return to sea level.
Given that , determine the value of .
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The path of a swing boat fairground ride that swings forwards and backwards is modelled as the arc of a circle, radius , as shown in the diagram below.
Ground level is represented by the -axis.
The value of represents the horizontal displacement, in metres, of the swing boat relative to the origin.
The value of represents the height, in metres, of the swing boat above ground level.
The height of the swing boat is modelled using
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The height, m, of water in a reservoir is modelled by the function
where is the time in hours after midday.
is a positive constant.
Write down the height of the water in the reservoir at midday.
The minimum height the water is m.
Find the height of the water at
giving your answers to two decimal places.
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A Ferris wheel is modelled as a circle with centre and radius m.
There are 32 passenger “pods” which are evenly spaced around the Ferris wheel.
A pod’s position can be determined by the angle, radians, which is measured anticlockwise from the positive -direction, as shown in the diagram below.
The coordinates of a pod, , are given by .
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As part of a quality control test, a lifejacket is thrown into the sea.
The height, m, of the lifejacket above or below sea level , at time seconds after first hitting the water, is modelled by the equation sin
The lifejacket reaches its furthest point below sea level after 0.64 seconds.
Find the distance below sea level this is, giving your answer to three significant figures.
According to the model, what should happen to the lifejacket as time progresses?
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The number of daylight hours, , is modelled using the function
sin
where is the day number on which the model applies.
Explain why the model does not quite cover a whole year before repeating itself.
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The alternating voltage,, in an electrical circuit, seconds after it is switched on, is modelled by the function
cos sin
where is measured in seconds.
Use the identity coscos cos sin sin to show that
cos sin
can be written as
After how many seconds does the voltage first equal -20 volts?
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A small spring is extended to its maximum length and released from rest.
The length of the spring, cm , at time seconds, where the angle is given in radians, is then modelled by the function
cos
State one criticism of this model as time passes.
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A dolphin is swimming such that it is diving in and out of the water at a constant speed. On each jump and dive the dolphin reaches a height of 2 m above sea level and a depth of 2 m below sea level.
Starting at sea level, the dolphin takes seconds to jump out of the water, dive back in and return to sea level.
Write down a model for the height, m, of the dolphin, relative to sea level, at time seconds, in the form sin where are constants to be found.
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The path of a swing boat fairground ride that swings forwards and backwards is modelled as a semi-circle, radius cm, as shown in the diagram below.
Ground level is represented by the -axis and represents the height of the boat above ground level. The path of the boat is given by the formula
The boat’s initial position is at the point .
Given that the -coordinate of the boat is also given by
sin
where seconds is the time since the boat was released from its initial position, find the time it takes the boat to swing from one end of the ride to the other.
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The height, m , of water in a reservoir is modelled by the function
where is the time in hours after midnight. and are positive constants.
In terms of and , write down the natural height of the water in the reservoir, as well as its maximum and minimum heights.
The maximum level of water is 3m higher than its natural level.
The level of water is three times higher at its maximum than at its minimum.
Find the maximum, minimum and natural water levels.
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A Ferris wheel with passenger “pods” is modelled as a circle with centre and radius m. A pod’s position can be determined by the angle, radians, which is measured anticlockwise from the positive -direction, as shown in the diagram below.
The coordinates of a pod, , are given by , where is a positive constant. Ground level is represented by the line with equation y=-60.
Find, to three significant figures, the angle for a passenger pod located at the point .
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A lifejacket falls over the side of a boat from a height of m.
The height, m, of the lifejacket above or below sea level , at time seconds after falling, is modelled by the equation cos .
The lifejacket reaches its furthest point below sea level after 0.742 seconds.
Find the total distance it has fallen, giving your answer to three significant figures.
Write down the value of for the first three times the lifejacket is at sea level.
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The number of daylight hours, , in the UK, during a day days after the spring equinox (the day in spring when the number of daylight hours is 12), is modelled using the function
For how many days of the year does the model suggest that the number of daylight hours exceeds 15 hours? Give your answer as a whole number of days.
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The alternating voltage, , in an electrical circuit, seconds after it is switched on, is modelled by the function
Show that
can be written as
sin
where and .
After how many seconds does the voltage first equal -55 volts?
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The length of a spring, cm , at time seconds, after being released from rest, is modelled by the function
cos
Describe what the constants and represent in terms of the length of the spring.
Given that the minimum length the spring can attain is cm and its maximum length is 30 cm, find the values of and .
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A hovering helicopter moves up and down at a constant rate between the heights of 200 m and 220 m. It takes the helicopter seconds to move between these two heights.
Write down a model in the form cos for the height, m, of the helicopter at time seconds, where are constants to be found.
State the initial height of the helicopter suggested by your model.
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The path of a swing boat fairground ride that swings forwards and backwards is modelled as a semi-circle, radius m, as shown in the diagram below.
At time seconds, the -coordinate of the boat is modelled by the function
sin
and the height, m , of the boat above the ground, at time seconds, is modelled by
Verify that the initial position of the boat is .
Find the position of the boat when it has swung through an angle of anticlockwise from the -axis, as shown in the diagram above.
Find the time at which the boat first reaches this position.
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The height, m, of water in a reservoir is modelled by the function
sin
where is the time, in hours, after midnight. and are positive constants.
The reservoir is only capable of holding water to a maximum height of m
Should the water level exceed this, an overflow reservoir is available.
During which times of day will the overflow reservoir be in use?
Give your answers to the nearest minute.
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A Ferris wheel with passenger “pods” is modelled as a circle with centre and radius m. A pod’s position can be determined by the angle radians, which is measured anticlockwise from the positive -direction, as shown in the diagram below.
The coordinates of a pod, are given by where and are positive constants. Ground level is represented by the line with equation y=-62.
Find the height above the ground of a passenger pod when radians.
Find the angle , to three significant figures, for a passenger pod located at the point .
What would you be able to say about the Ferris wheel in the case where ?
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A lifejacket falls over the side of a boat from a height of m above sea level.
The height,m , of the lifejacket above or below sea level at time seconds after falling, is modelled by the equation cos , where are positive constants.
After 2.054 seconds the lifejacket is m below sea level. Find the value of and determine whether the lifejacket is rising or sinking.
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The number of daylight hours, , in the UK, days after the spring equinox (the day in spring when the number of daylight hours is 12) is modelled using the function
sin
where are constants.
For how many days of the year does the number of daylight hours remain below 10? Give your answer as a whole number of days.
If the spring equinox falls on the 21st March, find the dates throughout the year when there are 16 hours of daylight.
The model needs to be adjusted every four years. Suggest a reason why.
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The alternating voltage, , in an electrical circuit seconds after it is switched on is modelled by the function
Express
in the form
sin
where are constants to be found. and is acute.
Find the voltage when the circuit is switched on.
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The length of a spring, cm, at time seconds, after being released from rest, is modelled by the function
cos
where and are constants.
Explain why the function would not be appropriate for modelling the length of a spring if .
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The height above ground,m , of a drone used as part of an air display is modelled by the function sin , where is the time in seconds after launch and are constants.
The drone is launched upwards from a height of 23 m and seconds later it reaches its maximum height of 26 m. The minimum height the drone reaches is 14 m.
Find the value of the constants ,,and given that is acute.
The drone’s lights switch off when its height drops below 17 m.
Show that the drone’s lights are on for two-thirds of its flight.
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A swing boat fairground ride is modelled as moving forwards and backwards along the path of a semi-circle, radius m, as shown in the diagram below.
Show that, for ,
The model is refined so that the coordinates of the boat can be calculated from the time, seconds, after the boat is set in motion. The and coordinates are now given by
sin
where is a constant.
For , find the times when the boat is equidistant from the ground and horizontally from the origin.
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The height of water, m , in a reservoir is modelled by the function
sin
where is the time in hours after midnight. and are positive constants.
Briefly explain how each of the constants and affect the height of the water in the reservoir.
Show that the height of water will first be at its minimum level at time
hours after midnight.
Show that the rate of change of the height of water in the reservoir is at its greatest every
hours after midnight.
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Engineers are designing a Ferris wheel with passenger “pods”.
The wheel is modelled as a circle with centre and radius meters.
One of the pods is to be located at the point with coordinates .
The thick lines on the diagram represent two symmetrical ground supports for the Ferris wheel each going from its centre to ground level.
The left-hand support is represented by the equation .
The -axis represents ground level.
The pods are to be evenly distributed around the wheel.
Ideally the engineers would like no more than three pods to be within the intersection of the supports at any one time. Find the maximum value of this design approach allows.
For both strength and aesthetic reasons, both the ground supports will be made in two sections. Thinner materials will be used within the wheel so as not to obstruct the view of, and from, the Ferris wheel and thicker material will be used for the lower base supports outside the wheel.
Find the percentage of the thicker material required.
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The height, m, of a helicopter, seconds after take-off, is modelled by the function
tan
The time lag between the pilot firing up the helicopter and leaving the ground is accounted for in the model by negative values of for the period .
Find the value of α to two significant figures.
Show that the helicopter rises just 4 m between the times of seconds and seconds
Find the height of the helicopter at the point at which the model ceases to be valid.
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The number of daylight hours, , in the UK, days after the spring equinox (the day in spring when the number of daylight hours is 12) is modelled using the function
where and are constants.
Explain the meaning of the constants and in the context of this model.
During a normal year (not a leap year), the maximum number of daylight hours is
16 hours and 38 minutes.
Find the total number of daylight hours in the first half of the year.
(Assume a year in this sense starts on the spring equinox, when .
Give your answer to the nearest 10 hours.
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The alternating voltage, , in a domestic electrical circuit, seconds after it is switched on is modelled by the function
sin cos
Express
sin cos
in the form
sin
where and are constants to be found. and is α acute.
In the UK, domestic electricity runs at a frequency, , of 50 Hertz (Hz).
The constant , is given by .
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