Applied context modeling

7. (a) Express \(\cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. A scientist is studying the behaviour of seabirds in a colony. She models the height above sea level, \(H\) metres, of one of the birds in the colony by the equation $$H = \frac { 24 } { 3 + \cos \left( \frac { 1 } { 2 } t \right) + 4 \sin \left( \frac { 1 } { 2 } t \right) } \quad 0 \leqslant t \leqslant 6.5$$ where \(t\) seconds is the time after it leaves the nest. Find, according to the model,
(b) the minimum height of the seabird above sea level, giving your answer to the nearest cm,
(c) the value of \(t\), to 2 decimal places, when \(H = 10\) \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-21_2255_50_314_34}

1. (a) Express \(2 \sin x - 4 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 significant figures. In a town in Norway, a student records the number of hours of daylight every day for a year. He models the number of hours of daylight, \(H\), by the continuous function given by the formula $$H = 12 + 4 \sin \left( \frac { 2 \pi t } { 365 } \right) - 8 \cos \left( \frac { 2 \pi t } { 365 } \right) , \quad 0 \leqslant t \leqslant 365$$ where \(t\) is the number of days since he began recording.
(b) Using your answer to part (a), or otherwise, find the maximum and minimum number of hours of daylight given by this formula. Give your answers to 3 significant figures.
(c) Use the formula to find the values of \(t\) when \(H = 17\), giving your answers to the nearest integer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

13. (a) Express \(2 \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to 2 decimal places.
\begin{figure}[h] \end{figure} Figure 4 shows the design for a logo that is to be displayed on the side of a large building. The logo consists of three rectangles, \(C , D\) and \(E\), each of which is in contact with two horizontal parallel lines \(l _ { 1 }\) and \(l _ { 2 }\). Rectangle \(D\) touches rectangles \(C\) and \(E\) as shown in Figure 4. Rectangles \(C , D\) and \(E\) each have length 4 m and width 2 m . The acute angle \(\theta\) between the line \(l _ { 2 }\) and the longer edge of each rectangle is shown in Figure 4. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are 4 m apart,
(b) show that $$2 \sin \theta + \cos \theta = 2$$ Given also that \(0 < \theta < 45 ^ { \circ }\),
(c) solve the equation $$2 \sin \theta + \cos \theta = 2$$ giving the value of \(\theta\) to 1 decimal place. Rectangles \(C\) and \(D\) and rectangles \(D\) and \(E\) touch for a distance \(h \mathrm {~m}\) as shown in Figure 4. Using your answer to part (c), or otherwise,
(d) find the value of \(h\), giving your answer to 2 significant figures.

11. (a) Express \(1.5 \sin \theta - 1.2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the value of \(R\) and the value of \(\alpha\) to 3 decimal places. The height, \(H\) metres, of sea water at the entrance to a harbour on a particular day, is modelled by the equation $$H = 3 + 1.5 \sin \left( \frac { \pi t } { 6 } \right) - 1.2 \cos \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
(b) Using your answer to part (a), calculate the minimum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this minimum occurs.
(c) Find, to the nearest minute, the times when the height of sea water at the entrance to the harbour is predicted by this model to be 4 metres.

1. (a) Write \(2 \sin \theta - \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha \leqslant 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to one decimal place.
   (3)

\begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph with equation \(y = 2 \sin \theta - \cos \theta , \quad 0 \leqslant \theta < 360 ^ { \circ }\) (b) Sketch the graph with equation $$y = | 2 \sin \theta - \cos \theta | , \quad 0 \leqslant \theta < 360 ^ { \circ }$$ stating the coordinates of all points at which the graph meets or cuts the coordinate axes. The temperature of a warehouse is modelled by the equation $$f ( t ) = 5 + \left| 2 \sin ( 15 t ) ^ { \circ } - \cos ( 15 t ) ^ { \circ } \right| , \quad 0 \leqslant t < 24$$ where \(\mathrm { f } ( t )\) is the temperature of the warehouse in degrees Celsius and \(t\) is the time measured in hours from midnight. State
(c) (i) the maximum value of \(f ( t )\),
(ii) the largest value of \(t\), for \(0 \leqslant t < 24\), at which this maximum value occurs. Give your answer to one decimal place.

7. (a) Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 4 decimal places. The height of sea water, \(H\) metres, on a harbour wall is modelled by the equation $$H = 6 + 2.5 \cos \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \sin \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
(b) Calculate the times at which the model predicts that the height of sea water on the harbour wall will be 4.6 metres. Give your answers to the nearest minute. \includegraphics[max width=\textwidth, alt={}, center]{a9870c94-0910-46ec-a54a-44a431cb324e-18_2257_54_314_1977}

8. (a) Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence find the maximum value of \(3 \cos \theta + 4 \sin \theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. The temperature, \(\mathrm { f } ( t )\), of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
(c) Calculate the minimum temperature of the warehouse as given by this model.
(d) Find the value of \(t\) when this minimum temperature occurs.

7. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 4 decimal places.
(b) (i) Find the maximum value of \(2 \sin \theta - 1.5 \cos \theta\).
(ii) Find the value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum occurs. Tom models the height of sea water, \(H\) metres, on a particular day by the equation $$H = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) hours is the number of hours after midday.
(c) Calculate the maximum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this maximum occurs.
(d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.

8. (a) Express \(9 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 4 decimal places.
(b) (i) State the maximum value of \(9 \cos \theta - 2 \sin \theta\) (ii) Find the value of \(\theta\), for \(0 < \theta < 2 \pi\), at which this maximum occurs. Ruth models the height \(H\) above the ground of a passenger on a Ferris wheel by the equation $$H = 10 - 9 \cos \left( \frac { \pi t } { 5 } \right) + 2 \sin \left( \frac { \pi t } { 5 } \right)$$ where \(H\) is measured in metres and \(t\) is the time in minutes after the wheel starts turning. \includegraphics[max width=\textwidth, alt={}, center]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-14_572_458_719_1158}
(c) Calculate the maximum value of \(H\) predicted by this model, and the value of \(t\), when this maximum first occurs. Give your answers to 2 decimal places.
(d) Determine the time for the Ferris wheel to complete two revolutions.

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\), with equation \(y = 6 \cos x + 2.5 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\)

1. Express \(6 \cos x + 2.5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) to 3 decimal places.
2. Find the coordinates of the points on the graph where the curve \(C\) crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by $$H = 12 + 6 \cos \left( \frac { 2 \pi t } { 52 } \right) + 2.5 \sin \left( \frac { 2 \pi t } { 52 } \right) , \quad 0 \leqslant t \leqslant 52$$ where \(H\) is the number of hours of daylight and \(t\) is the number of weeks since her first recording. Use this function to find
3. the maximum and minimum values of \(H\) predicted by the model,
4. the values for \(t\) when \(H = 16\), giving your answers to the nearest whole number.
   [0pt] [You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.] \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-14_40_58_2460_1893}

8 The height of tide at the entrance to a harbour on a particular day may be modelled by the function \(h = 3 + 2 \sin 30 t + 1.5 \cos 30 t\) where \(h\) is measured in metres, \(t\) in hours after midnight and \(30 t\) is in degrees.
[0pt] [The values 2 and 1.5 represent the relative effects of the moon and sun respectively.]

1. Show that \(2 \sin 30 t + 1.5 \cos 30 t\) can be written in the form \(2.5 \sin ( 30 t + \alpha )\), where \(\alpha\) is to be determined.
2. Find the height of tide at high water and the first time that this occurs after midnight.
3. Find the range of tide during the day.
4. Sketch the graph of \(h\) against \(t\) for \(0 \leq t \leq 12\), indicating the maximum and minimum points.
5. A sailing boat may enter the harbour only if there is at least 2 metres of water. Find the times during this morning when it may enter the harbour.
6. From your graph estimate the time at which the water falling fastest and the rate at which it is falling.

1. (a) Express \(140 \cos \theta - 480 \sin \theta\) in the form \(K \cos ( \theta + \alpha )\) where \(K > 0\) and \(0 < \alpha < 90 ^ { \circ }\) State the value of \(K\) and give the value of \(\alpha\), in degrees, to 2 decimal places.

A scientist studies the number of rabbits and the number of foxes in a wood for one year. The number of rabbits, \(R\), is modelled by the equation $$R = A + 140 \cos ( 30 t ) ^ { \circ } - 480 \sin ( 30 t ) ^ { \circ }$$ where \(t\) months is the time after the start of the year and \(A\) is a constant.
Given that, during the year, the maximum number of rabbits in the wood is 1500
(b) (i) find a complete equation for this model.
(ii) Hence write down the minimum number of rabbits in the wood during the year according to the model. The actual number of rabbits in the wood is at its minimum value in the middle of April.
(c) Use this information to comment on the model for the number of rabbits. The number of foxes, \(F\), in the wood during the same year is modelled by the equation $$F = 100 + 70 \sin ( 30 t + 70 ) ^ { \circ }$$ The number of foxes is at its minimum value after \(T\) months.
(d) Find, according to the models, the number of rabbits in the wood at time \(T\) months.

1. (a) Express \(\sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.

The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside a room on a given day is modelled by the equation $$\theta = 5 + \sin \left( \frac { \pi t } { 12 } - 3 \right) + 2 \cos \left( \frac { \pi t } { 12 } - 3 \right) \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
Using the equation of the model and your answer to part (a),
(b) deduce the maximum temperature of the room during this day,
(c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

13. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places. Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula $$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017. Use Tom's model to
(b) find the depth of water at 00:00 hours on 18th October 2017,
(c) find the maximum depth of water,
(d) find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula $$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$ where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
(e) (i) explain why Jolene's model is not correct,
(ii) hence find a suitable model for \(H\) in terms of \(x\).

1. (a) Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. \begin{figure}[h] \end{figure} Figure 6 shows the cross-section of a water wheel.
The wheel is free to rotate about a fixed axis through the point \(C\).
The point \(P\) is at the end of one of the paddles of the wheel, as shown in Figure 6.
The water level is assumed to be horizontal and of constant height.
The vertical height, \(H\) metres, of \(P\) above the water level is modelled by the equation $$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$ where \(t\) is the time in seconds after the wheel starts rotating.
Using the model, find
(b) (i) the maximum height of \(P\) above the water level,
(ii) the value of \(t\) when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, \(P\) is below the water level for a total of \(T\) seconds. According to the model,
(c) find the value of \(T\) giving your answer to 3 significant figures.
(Solutions based entirely on calculator technology are not acceptable.) In reality, the water level may not be of constant height.
(d) Explain how the equation of the model should be refined to take this into account.

13. (a) Express \(10 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the exact value of \(R\) and give the value of \(\alpha\), in degrees, to 2 decimal places. \begin{figure}[h] \end{figure} The height above the ground, \(H\) metres, of a passenger on a Ferris wheel \(t\) minutes after the wheel starts turning, is modelled by the equation $$H = a - 10 \cos ( 80 t ) ^ { \circ } + 3 \sin ( 80 t ) ^ { \circ }$$ where \(a\) is a constant.
Figure 3 shows the graph of \(H\) against \(t\) for two complete cycles of the wheel.
Given that the initial height of the passenger above the ground is 1 metre,
(b) (i) find a complete equation for the model,
(ii) hence find the maximum height of the passenger above the ground.
(c) Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle.
(Solutions based entirely on graphical or numerical methods are not acceptable.) It is decided that, to increase profits, the speed of the wheel is to be increased.
(d) How would you adapt the equation of the model to reflect this increase in speed?

14 Douglas wants to construct a model for the height of the tide in Liverpool during the day, using a cosine graph to represent the way the height changes. He knows that the first high tide of the day measures 8.55 m and the first low tide of the day measures 1.75 m . Douglas uses \(t\) for time and \(h\) for the height of the tide in metres. With his graph-drawing software set to degrees, he begins by drawing the graph of \(\mathrm { h } = 5.15 + 3.4\) cost.

1. Verify that this equation gives the correct values of \(h\) for the high and low tide. Douglas also knows that the first high tide of the day occurs at 1 am and the first low tide occurs at 7.20 am. He wants \(t\) to represent the time in hours after midnight, so he modifies his equation to \(h = 5.15 + 3.4 \cos ( a t + b )\).
2. 1. Show that Douglas's modified equation gives the first high tide of the day occurring at the correct time if \(\mathrm { a } + \mathrm { b } = 0\).
   2. Use the time of the first low tide of the day to form a second equation relating \(a\) and \(b\).
   3. Hence show that \(a = 28.42\) correct to 2 decimal places.
3. Douglas can only sail his boat when the height of the tide is at least 3 m . Use the model to predict the range of times that morning when he cannot sail.
4. The next high tide occurs at 12.59 pm when the height of the tide is 8.91 m . Comment on the suitability of Douglas's model.

1. (a) Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).

Give the values of \(R\) and \(\alpha\) to 3 significant figures.
(b)Find the maximum and minimum values of \(2 \cos \theta + 5 \sin \theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. The temperature \(T ^ { \circ } \mathrm { C }\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac { \pi t } { 12 } \right) + 5 \sin \left( \frac { \pi t } { 12 } \right) , \quad 0 \leq t < 24$$ where \(t\) hours is the number of hours after 1200 .
(c) Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs.
(d) Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 ^ { \circ } \mathrm { C }\).

1. During 2029, the number of hours of daylight per day in London, H, is modelled by the equation

$$H = 0.3 \sin \left( \frac { x } { 60 } \right) - 4 \cos \left( \frac { x } { 60 } \right) + 11.5 \quad 0 \leqslant x < 365$$ where \(x\) is the number of days after 1st January 2029 and the angle is in radians.

1. Show that, according to the model, the number of hours of daylight in London on the 31st January 2029 will be 8.13 to 3 significant figures.
2. Use the substitution \(t = \tan \left( \frac { x } { 120 } \right)\) to show that \(H\) can be written as $$H = \frac { a t ^ { 2 } + b t + c } { 1 + t ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are constants to be determined.
3. Hence determine, according to the model, the date of the first day of 2029 when there will be at least 12 hours of daylight in London.

6. (a) Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the values of \(R\) and \(\alpha\) to 3 significant figures.
(b) Find the maximum and minimum values of \(2 \cos \theta + 5 \sin \theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. The temperature \(T ^ { \circ } \mathrm { C }\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac { \pi t } { 12 } \right) + 5 \sin \left( \frac { \pi t } { 12 } \right) , \quad 0 \leq t < 24 ,$$ where \(t\) hours is the number of hours after 1200 .
(c) Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs.
(d) Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 ^ { \circ } \mathrm { C }\).