1.08k Separable differential equations: dy/dx = f(x)g(y)

8

1. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ given that \(x = 20\) when \(t = \frac { \pi } { 4 }\), giving your solution in the form \(x ^ { 2 } = \mathrm { f } ( t )\). (6 marks)
2. The oscillations of a 'baby bouncy cradle' are modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ where \(x \mathrm {~cm}\) is the height of the cradle above its base \(t\) seconds after the cradle begins to oscillate. Given that the cradle is 20 cm above its base at time \(t = \frac { \pi } { 4 }\) seconds, find:
   1. the height of the cradle above its base 13 seconds after it starts oscillating, giving your answer to the nearest millimetre;
   2. the time at which the cradle will first be 11 cm above its base, giving your answer to the nearest tenth of a second.
      (2 marks)

8 Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population \(x\), in thousands, of red squirrels is modelled by the equation $$x = \frac { a } { 1 + k t } ,$$ where \(t\) is the time in years, and \(a\) and \(k\) are constants. When \(t = 0 , x = 2.5\).

1. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { k x ^ { 2 } } { a }\).
2. Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate \(a\) and \(k\).
3. What is the long-term population of red squirrels predicted by this model? The population \(y\), in thousands, of grey squirrels is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 2 y - y ^ { 2 } .$$ When \(t = 0 , y = 1\).
4. Express \(\frac { 1 } { 2 y - y ^ { 2 } }\) in partial fractions.
5. Hence show by integration that \(\ln \left( \frac { y } { 2 - y } \right) = 2 t\). Show that \(y = \frac { 2 } { 1 + \mathrm { e } ^ { - 2 t } }\).
6. What is the long-term population of grey squirrels predicted by this model? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4)
   Paper B: Comprehension
   Monday
   23 JANUARY 2006
   Afternooon U Additional materials:
   Rough paper
   MEI Examination Formulae and Tables (MF2)
   4754(B) \section*{Up to 1 hour $$\text { o to } 1 \text { hour }$$} \section*{TIME} \section*{Up to 1 hour}

   For Examiner's Use
   Qu.	Mark
   1
   2
   3
   4
   5
   Total

   1 Line 59 says "Again Party G just misses out; if there had been 7 seats G would have got the last one." Where is the evidence for this in the article? 26 parties, P, Q, R, S, T and U take part in an election for 7 seats. Their results are shown in the table below.

   Party	Votes (\%)
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3

1. Use the Trial-and-Improvement method, starting with values of \(10 \%\) and \(14 \%\), to find an acceptance percentage for 7 seats, and state the allocation of the seats.

   Acceptance percentage, \(\boldsymbol { a }\) \%		10\%	14\%
   Party	Votes (\%)	Seats	Seats	Seats	Seats	Seats
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3
   Total seats

   Seat Allocation \(\quad \mathrm { P } \ldots\)... \(\mathrm { Q } \ldots\) R ... S ... T ... \(\mathrm { U } \ldots\).
2. Now apply the d'Hondt Formula to the same figures to find the allocation of the seats.

   	Round
   Party	1	2	3	4	5	6	7	Residual
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3
   Seat allocated to

   Seat Allocation \(\mathrm { P } \ldots\). Q ... \(\mathrm { R } \ldots\). S ... T ... \(\mathrm { U } \ldots\). 3 In this question, use the figures for the example used in Table 5 in the article, the notation described in the section "Equivalence of the two methods" and the value of 11 found for \(a\) in Table 4. Treating Party E as Party 5, verify that \(\frac { V _ { 5 } } { N _ { 5 } + 1 } < a \leqslant \frac { V _ { 5 } } { N _ { 5 } }\).
   4 Some of the intervals illustrated by the lines in the graph in Fig. 8 are given in this table.

   Seats	Interval	Seats	Interval
   1	\(22.2 < a \leqslant 27.0\)	5
   2	\(16.6 < a \leqslant 22.2\)	6	\(10.6 < a \leqslant 11.1\)
   3		7
   4

1. Describe briefly, giving an example, the relationship between the end-points of these intervals and the values in Table 5, which is reproduced below.
2. Complete the table above. \begin{table}[h]

   	Round
   Party	1	2	3	4	5	6	Residual
   A	22.2	22.2	11.1	11.1	11.1	11.1	7.4
   B	6.1	6.1	6.1	6.1	6.1	6.1	6.1
   C	27.0	13.5	13.5	13.5	9.0	9.0	9.0
   D	16.6	16.6	16.6	8.3	8.3	8.3	8.3
   E	11.2	11.2	11.2	11.2	11.2	5.6	5.6
   F	3.7	3.7	3.7	3.7	3.7	3.7	3.7
   G	10.6	10.6	10.6	10.6	10.6	10.6	10.6
   H	2.6	2.6	2.6	2.6	2.6	2.6	2.6
   Seat allocated to	C	A	D	C	E	A

   \captionsetup{labelformat=empty} \caption{Table 5}
   \end{table} 5 The ends of the vertical lines in Fig. 8 are marked with circles. Those at the tops of the lines are filled in, e.g. • whereas those at the bottom are not, e.g. o.

1. Relate this distinction to the use of inequality signs.
2. Show that the inequality on line 102 can be rearranged to give \(0 \leqslant V _ { k } - N _ { k } a < a\). [1]
3. Hence justify the use of the inequality signs in line 102.

4

1. The number of bacteria in a colony is increasing at a rate that is proportional to the square root of the number of bacteria present. Form a differential equation relating \(x\), the number of bacteria, to the time \(t\).
2. In another colony, the number of bacteria, \(y\), after time \(t\) minutes is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = \frac { 10000 } { \sqrt { y } } .$$ Find \(y\) in terms of \(t\), given that \(y = 900\) when \(t = 0\). Hence find the number of bacteria after 10 minutes.

8 A stone, of mass 0.05 kg , is moving along the smooth horizontal floor of a tank, which is filled with oil. At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a resistance force of magnitude \(0.08 v ^ { 2 }\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 1.6 v ^ { 2 }$$ (2 marks)
2. The initial speed of the stone is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that $$v = \frac { 15 } { 5 + 24 t }$$ (5 marks)

9 A bungee jumper, of mass 80 kg , is attached to one end of a light elastic cord, of natural length 16 metres and modulus of elasticity 784 N . The other end of the cord is attached to a horizontal platform, which is at a height of 65 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached and falls vertically. The bungee jumper can be modelled as a particle. Hooke's law can be assumed to apply throughout the motion and air resistance can be assumed to be negligible.

1. Find the length of the cord when the acceleration of the bungee jumper is zero.
2. The cord extends by \(x\) metres beyond its natural length before the bungee jumper first comes to rest.
   1. Show that \(x ^ { 2 } - 32 x - 512 = 0\).
   2. Find the distance above the ground at which the bungee jumper first comes to rest.

5 A golf ball, of mass \(m \mathrm {~kg}\), is moving in a straight line across smooth horizontal ground. At time \(t\) seconds, the golf ball has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the golf ball moves, it experiences a resistance force of magnitude \(0.2 m v ^ { \frac { 1 } { 2 } }\) newtons until it comes to rest. No other horizontal force acts on the golf ball. Model the golf ball as a particle.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.2 v ^ { \frac { 1 } { 2 } }$$
2. When \(t = 0\), the speed of the golf ball is \(16 \mathrm {~ms} ^ { - 1 }\). Show that \(v = ( 4 - 0.1 t ) ^ { 2 }\).
3. Find the value of \(t\) when \(v = 1\).
4. Find the distance travelled by the golf ball as its speed decreases from \(16 \mathrm {~ms} ^ { - 1 }\) to \(1 \mathrm {~ms} ^ { - 1 }\).

6 A car, of mass \(m\), is moving along a straight smooth horizontal road. At time \(t\), the car has speed \(v\). As the car moves, it experiences a resistance force of magnitude \(0.05 m v\). No other horizontal force acts on the car.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.05 v$$
2. When \(t = 0\), the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(v = 20 \mathrm { e } ^ { - 0.05 t }\).
3. Find the time taken for the speed of the car to reduce to \(10 \mathrm {~ms} ^ { - 1 }\).

8 A stone, of mass \(m\), is moving in a straight line along smooth horizontal ground.
At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a total resistance force of magnitude \(\lambda m v ^ { \frac { 3 } { 2 } }\), where \(\lambda\) is a constant. No other horizontal force acts on the stone.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v ^ { \frac { 3 } { 2 } }$$ (2 marks)
2. The initial speed of the stone is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that $$v = \frac { 36 } { ( 2 + 3 \lambda t ) ^ { 2 } }$$ (7 marks)
3. Find, in terms of \(\lambda\), the time taken for the speed of the stone to drop to \(4 \mathrm {~ms} ^ { - 1 }\).

2. At time \(t = 0\), a particle \(P\) of mass \(m\) is projected vertically upwards with speed \(\sqrt { \frac { g } { k } }\), where \(k\) is a constant. At time \(t\) the speed of \(P\) is \(v\). The particle \(P\) moves against air resistance whose magnitude is modelled as being \(m k v ^ { 2 }\) when the speed of \(P\) is \(v\). Find, in terms of \(k\), the distance travelled by \(P\) until its speed first becomes half of its initial speed.

13 In this question you must show detailed reasoning. Find the exact values of the \(x\)-coordinates of the stationary points of the curve \(x ^ { 3 } + y ^ { 3 } = 3 x y + 35\).

14 John wants to encourage more birds to come into the park near his house. Each day, starting on day 1, he puts bird food out and then observes the birds for one hour. He records the maximum number of birds that he observes at any given moment in the park each day. He believes that his observations may be modelled by the following differential equation, where \(n\) is the maximum number of birds that he observed at any given moment on day \(t\). \(\frac { \mathrm { d } n } { \mathrm {~d} t } = 0.1 n \left( 1 - \frac { n } { 50 } \right)\)

1. Show that the general solution to the differential equation can be written in the form \(n = \frac { 50 A } { \mathrm { e } ^ { - 0.1 t } + A }\), where \(A\) is an arbitrary positive constant.
2. Using his model, determine the maximum number of birds that John would expect to observe at any given moment in the long term.
3. Write down one possible refinement of this model.
4. Write down one way in which John's model is not appropriate. \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
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   3(a) \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-27_801_1479_214_360} 3(b) \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-27_1150_1504_1452_338}

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8 \includegraphics[max width=\textwidth, alt={}, center]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-07_360_489_1027_788} The diagram shows a water tank which is shaped as an inverted cone with semi-vertical angle \(30 ^ { \circ }\) and height 50 cm . Initially the tank is full, and the depth of the water is 50 cm . Water flows out of a small hole at the bottom of the tank. The rate at which the water flows out is modelled by \(\frac { \mathrm { d } V } { \mathrm {~d} t } = - 2 h\), where \(V \mathrm {~cm} ^ { 3 }\) is the volume of water remaining and \(h \mathrm {~cm}\) is the depth of water in the tank \(t\) seconds after the water begins to flow out. Determine the time taken for the tank to become empty.
[0pt] [For a cone with base radius \(r\) and height \(h\) the volume \(V\) is given by \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

11 A curve, \(C\), passes through the point with coordinates \(( 1,6 )\) The gradient of \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 6 } ( x y ) ^ { 2 }$$ Show that \(C\) intersects the coordinate axes at exactly one point and state the coordinates of this point. Fully justify your answer.

16

1. Given that $$\frac { 1 } { 16 - 9 x ^ { 2 } } \equiv \frac { A } { 4 - 3 x } + \frac { B } { 4 + 3 x }$$ find the values of \(A\) and \(B\) 16
2. An empty container, in the shape of a cuboid, has length 1.6 metres, width 1.25 metres and depth 0.5 metres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-29_469_812_404_616} The container has a small hole in the bottom. Water is poured into the container at a rate of 0.16 cubic metres per minute.
   At time \(t\) minutes after the container starts to be filled, the depth of water is \(d\) metres and water leaks out at a rate of \(0.36 d ^ { 2 }\) cubic metres per minute. At time \(t\) minutes after the container starts to be filled, the volume of water in the container is \(V\) cubic metres. 16 (b) (i) Show that $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 16 - 9 V ^ { 2 } } { 100 }$$ \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-30_2493_1721_214_150} \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-31_2492_1721_217_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
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   Write the question numbers in the left-hand margin. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-36_2498_1723_213_148}

17 A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth. 17

1. Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds after being released from rest, is \(2 g \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show how Andy has obtained his prediction.
   17
2. Using a refined model, Amy predicts that the ball's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), at time \(t\) seconds after being released from rest is $$a = g - 0.1 v$$ where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the ball at time \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
   17
3. Comment on the value of \(v\) for the two models as \(t\) becomes large.

10 A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects.
Initially there are 25 damaged tomato plants.
The number of tomato plants damaged by insects is increasing by \(32 \%\) each day.
10

1. The total number of plants damaged by insects, \(x\), is modelled by $$x = A \times B ^ { t }$$ where \(A\) and \(B\) are constants and \(t\) is the number of days after the gardener first noticed the damaged plants. 10
   1. 1. Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.
         10
      2. Explain why this model is not realistic in the long term.
         10
      3. A refined model assumes the rate of increase of the number of plants damaged by insects is given by $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ( 900 - x ) } { 2700 }$$
      10
   2. 1. Show that $$\int \left( \frac { A } { x } + \frac { B } { 900 - x } \right) \mathrm { d } x = \int \mathrm { d } t$$ where \(A\) and \(B\) are positive integers to be found.
         

         10 (iii) Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

11 Solve the differential equation \(x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \sec y\) given that \(y = \frac { \pi } { 6 }\) when \(x = 4\) giving your answer in the form \(y = \mathrm { f } ( x )\).

10 A tank with vertical sides and rectangular cross-section is initially full of water. The water is leaking out of a hole in the base of the tank at a rate which is proportional to the square root of the depth of the water. \(V \mathrm {~m} ^ { 3 }\) is the volume of water in the tank at time \(t\) hours.

1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = a \sqrt { V }\), where \(a\) is a constant.
2. Given that the tank is half full after one hour, show that \(V = V _ { 0 } \left( \left( \frac { 1 } { \sqrt { 2 } } - 1 \right) t + 1 \right) ^ { 2 }\), where \(V _ { 0 } \mathrm {~m} ^ { 3 }\) is the initial volume of water in the tank.
3. Hence show that the tank will be empty after approximately 3 hours and 25 minutes.

13 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - k ( y - 10 )\), where \(k\) is a constant, given that \(y = 70\) when \(x = 0\) and \(y = 40\) when \(x = 1\). Express your answer in the form \(y = a + b \left( \frac { 1 } { 2 } \right) ^ { x }\) where \(a\) and \(b\) are constants to be found.
[0pt] [10]

12 A bullet of mass 0.0025 kg is fired vertically upwards from a point \(O\). At time \(t \mathrm {~s}\) after projection the speed of the bullet is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion has magnitude \(0.00001 v ^ { 2 } \mathrm {~N}\).

1. Show that, while the bullet is rising, $$250 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - 2500 - v ^ { 2 }$$
2. It is given that the speed of projection is \(350 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
   1. the time taken after projection for the bullet to reach its greatest height above \(O\),
   2. the greatest height above \(O\) reached by the bullet.

11 A differential equation is given by \(2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ( 1 - y )\).

1. Express \(\frac { 2 } { y ( 1 - y ) }\) in partial fractions.
2. Hence show by integration that \(\frac { y ^ { 2 } } { ( 1 - y ) ^ { 2 } } = A \mathrm { e } ^ { x }\).
3. Given that \(x = 0\) when \(y = 2\), find the value of \(A\) and express \(y\) in terms of \(x\).

6 A cup of tea is served at \(80 ^ { \circ } \mathrm { C }\) in a room which is kept at a constant \(20 ^ { \circ } \mathrm { C }\). The temperature, \(T ^ { \circ } \mathrm { C }\), of the tea after \(t\) minutes can be modelled by assuming that the rate of change of \(T\) is proportional to the difference in temperature between the tea and the room.

1. Explain why the rate of change of the temperature in this model is given by \(\frac { \mathrm { d } T } { \mathrm {~d} t } = - k ( T - 20 )\), where \(k\) is a positive constant.
2. Show by integration that the temperature of the tea after \(t\) minutes is given by \(T = 20 + 60 \mathrm { e } ^ { - k t }\).
3. After 2 minutes the tea has cooled to \(60 ^ { \circ } \mathrm { C }\). Find the value of \(k\).

10 A small body of mass \(m\) is thrown vertically upwards with initial velocity \(u\). Resistance to motion is \(k v ^ { 2 }\) per unit mass, where the velocity is \(v\) and \(k\) is a positive constant. Find, in terms of \(u , g\) and \(k\),

1. the time taken to reach the greatest height,
2. the greatest height to which the body will rise.

12 A patch of disease on a leaf is being chemically treated. At time \(t\) days after treatment starts, its length is \(x \mathrm {~cm}\) and the rate of decrease of its length is observed to be inversely proportional to the square root of its length. At time \(t = 3 , x = 4\) and, at this instant, the length is decreasing at 0.05 cm per day. Write down and solve a differential equation to model this situation. Hence find the time it takes for the length to decrease to 0.01 cm .
[0pt] [10]