A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 10: Numerical methods · Lesson 4

Applications to modelling

122 marks

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Quick questions to start.

A1
Needed today
Given $f (x) = x^{2} e^{- 0.5 x}$ , find $f^{'} (x)$ and the value of $f^{'} (2)$ to 4 decimal places.
[3 marks]
Answer
A2
Last lesson
Use the Newton–Raphson method on $f (x) = x^{3} - 10$ with $x_{0} = 2$ to find $x_{1}$ and $x_{2}$ to 4 decimal places.
[3 marks]
Answer
A3
Last lesson
Use $x_{n + 1} = 3 + ln x_{n}$ with $x_{0} = 4$ to find $x_{1}$ , $x_{2}$ and $x_{3}$ to 4 decimal places.
[2 marks]
Answer
A4
Common slip
A model gives a time of $t = 4.37$ hours. Write this time in hours and minutes, to the nearest minute.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

The value of a van, $£ V$ , $t$ years after it is bought, is modelled by $V = 18000 e^{- 0.2 t} - 400 t$ . (a) Find the value after $3$ years, to the nearest $£100$ . (b) Show that the model gives a value of zero between $t = 8$ and $t = 9$ . (c) Use the Newton–Raphson method with $t_{0} = 8.5$ to find $t_{1}$ to 3 decimal places. (d) Give one criticism of the model for large $t$ .

Hint Write

f (t) = 18000 e^{- 0.2 t} - 400 t

. Then

f^{'} (t) = - 3600 e^{- 0.2 t} - 400

. A value cannot be negative.

B1
The value of a bike, $£ B$ , $t$ years after it is bought, is modelled by $B = 3000 (0.78)^{t} - 150 t$ . Show that $B = 0$ between $t = 5$ and $t = 6$ . Use the Newton–Raphson method with $t_{0} = 5.5$ to find $t_{1}$ to 3 decimal places. Explain why the model is not valid for $t > t_{1}$ .
[6 marks]
Answer

Example 2

A runner's speed, $v$ m/s, $t$ seconds after the start is modelled by $v = (8 - 0.5 t) ln (t + 1)$ , $0 \leq t \leq 16$ . (a) Show that $\frac{d v}{d t} = 0$ can be written $t = \frac{17}{1 + ln ( t + 1 )} - 1$ . (b) Use $t_{n + 1} = \frac{17}{1 + ln ( t _{n} + 1 )} - 1$ with $t_{0} = 5$ to find the time of the greatest speed to 3 decimal places. (c) Find the greatest speed.

Hint Product rule:

\frac{d v}{d t} = - 0.5 ln (t + 1) + \frac{8 - 0.5 t}{t + 1}

. Multiply by

2 (t + 1)

, then add

t + 1

to both sides.

B2
The power output, $P$ kW, of a machine $t$ hours after it starts is modelled by $P = (12 - t) ln (t + 2)$ , $0 \leq t \leq 12$ . Show that $\frac{d P}{d t} = 0$ can be written $t = \frac{14}{1 + ln ( t + 2 )} - 2$ . Use this iteration with $t_{0} = 3$ to find the time of the greatest output to 3 decimal places, and find the greatest output.
[7 marks]
Answer

Example 3

A circular pond has radius $6$ m. A straight bridge cuts off a segment of the pond with area $20$ m². The angle at the centre is $θ$ radians. (a) Show that $θ - sin θ = \frac{10}{9}$ . (b) Use the Newton–Raphson method with $θ_{0} = 2$ to find $θ$ to 4 decimal places. (c) Find the length of the bridge.

Hint The area of a segment is

\frac{1}{2} r^{2} (θ - sin θ)

. The chord has length

2 r sin (\frac{θ}{2})

B3
A circular lawn has radius $5$ m. A straight path cuts off a segment of area $15$ m², with angle $θ$ radians at the centre. Show that $θ - sin θ = 1.2$ . Use the Newton–Raphson method with $θ_{0} = 2$ to find $θ$ to 3 decimal places, and find the length of the path.
[6 marks]
Answer

Example 4

The concentration, $C$ mg/l, of a painkiller in the blood $t$ hours after a dose is $C = 10 t e^{- 0.4 t}$ . It works while $C \geq 3$ . The concentration falls back to $3$ at time $T$ , where $T$ is near $8$ . Use the Newton–Raphson method with $t_{0} = 8$ to find $T$ , and show that the painkiller stops working at $8$ hours $18$ minutes, to the nearest minute.

Hint Use

f (t) = 10 t e^{- 0.4 t} - 3

and

f^{'} (t) = (10 - 4 t) e^{- 0.4 t}

8

18

min to the nearest minute means a time between

8

17.5

min and

8

18.5

min.

B4
The amount of a pollutant in a river, $P$ units, $t$ days after a spill is $P = 25 t e^{- 0.3 t}$ . The river is safe when $P < 5$ . The amount falls to $5$ at time $T$ days, where $T$ is near $14$ . Use the Newton–Raphson method with $t_{0} = 14$ to find $T$ to 4 decimal places. Show that the river becomes safe $14$ days $5$ hours after the spill, to the nearest hour.
[6 marks]
Answer

Practice

A-Level

C1
The depth of water in a harbour, $d$ metres, $t$ hours after midnight is modelled by $d = 5 + 2 sin (0.5 t) - 0.1 t$ , with the angle in radians. A boat needs a depth of at least $4$ m. Mia solves $f (t) = 1 + 2 sin (0.5 t) - 0.1 t = 0$ with the Newton–Raphson method and $t_{0} = 7$ . Her calculator is in degrees and she gets $t_{1} = 6.5300$ and $t_{2} = 6.0170$ . Explain her mistake. Show that the correct root lies between $6$ and $7$ , find the correct $t_{1}$ and $t_{2}$ to 4 decimal places, and give the time the depth first falls to $4$ m, to the nearest minute.
[7 marks]

C2
A tank starts filling at $09 : 00$ . The volume of water, $V$ litres, $t$ hours later is $V = 120 (1 - e^{- 0.3 t}) + 8 t$ . An alarm sounds when $V = 150$ . Ali finds the root $t = 6.1327$ and says the alarm sounds at $15 : 13$ . Explain Ali's mistake. Show that the alarm sounds at $15 : 08$ , to the nearest minute.
[5 marks]
C3
A section of a roller coaster track has height $h = 30 + 8 sin (0.4 x) - 0.05 x^{2}$ metres, $x$ metres along the ground from the start, $0 \leq x \leq 16$ . The track has a lowest point at $x = β$ , where $β$ is near $10$ . Find $\frac{d h}{d x}$ and $\frac{d ^{2} h}{d x ^{2}}$ . Apply the Newton–Raphson method to $\frac{d h}{d x} = 0$ with $x_{0} = 10$ to find $β$ to 3 decimal places. Find the height at this point and show that it is a minimum.
[7 marks]
Answer
C4
A ball is thrown upwards from a cliff top. Its height above the sea, $h$ metres, $t$ seconds later is modelled by $h = 2 + 14 t - 4.9 t^{2} - 0.5 e^{0.3 t}$ . Leo uses the Newton–Raphson method with $t_{0} = 0$ and gets $t_{2} = - 0.1044$ . He says the ball lands at $- 0.10$ s. Explain why his answer is not valid. Show that the ball lands between $t = 2.5$ and $t = 3$ , then use $t_{0} = 3$ to find the time it lands, to 2 decimal places.
[6 marks]

Exam-style questions

Edexcel exam style

D1
The concentration, $C$ mg/l, of a drug in a patient's blood $t$ hours after an injection is modelled by $C = 6 (e^{- 0.2 t} - e^{- 0.7 t})$ . The drug is effective while $C \geq 2$ .
(a)
Find $\frac{d C}{d t}$ and show that the concentration is greatest when $t = 2 ln 3.5$ .
[3 marks]
(b)
The concentration falls back to $2$ at time $T$ . Show that $T$ lies between $5$ and $5.5$ .
[2 marks]
(c)
Use the Newton–Raphson method with $t_{0} = 5.5$ to find $t_{1}$ and $t_{2}$ to 4 decimal places.
[3 marks]
Answer
(d)
The concentration first reaches $2$ at $t = 1.0631$ . Find, to the nearest minute, how long the drug is effective.
[2 marks]
Answer
Total for D1: 10 marks

D2
Jo models the temperature in a greenhouse, $T °$ C, $t$ hours after midnight by $T = 3 + 0.12 (t - 6)^{2} + ln (t + 1)$ , $0 \leq t \leq 12$ . She writes: " $T (0) = 7.32$ and $T (12) = 9.88$ . Both are above $5$ and there is no change of sign, so the temperature never falls below $5 °$ C."
Jo is wrong.
(a)
Explain why Jo's argument is not valid.
[2 marks]
(b)
By evaluating $T$ at $t = 4$ , $5$ and $7$ , show that the temperature falls below $5 °$ C and then rises above it.
[3 marks]
(c)
The two times are $4.3683$ and $6.3063$ hours after midnight. Find, to the nearest minute, how long the temperature is below $5 °$ C.
[3 marks]
Answer
Total for D2: 8 marks
D3
Kepler's equation links the position of a planet in its orbit to time: $M = E - 0.25 sin E$ , where the angles are in radians. When $M = 1.2$ , find $E$ correct to 3 decimal places. Use the Newton–Raphson method with $E_{0} = 1.2$ , and show that your answer is correct to this accuracy.
[7 marks]
Answer
D4
A water rocket is launched from the ground. Its height, $h$ metres, $t$ seconds after launch is modelled by $h = 30 t - 5 t^{2} - 2 (e^{0.4 t} - 1)$ , $t \geq 0$ . The rocket lands at time $T$ seconds.
(a)
Show that $T$ lies between $5$ and $5.5$ .
[2 marks]
(b)
Find $\frac{d h}{d t}$ .
[2 marks]
Answer
(c)
Use the Newton–Raphson method with $t_{0} = 5$ to find $T$ to 2 decimal places. Show that your answer is correct to this accuracy.
[3 marks]
Answer
(d)
The greatest height is reached at $t = 2.7588$ . Find the greatest height to 3 significant figures, and give one reason why the model may not be suitable.
[2 marks]
Total for D4: 9 marks
D5
A species of fish, A, lives in a lake. At $t = 0$ a second species, B, is introduced. The numbers of each, $t$ years later, are modelled by $A = 50 e^{0.12 t}$ and $B = 300 ln (t + 1)$ .
(a)
Show that $A = B$ for a value of $t$ between $24$ and $25$ .
[2 marks]
(b)
Use the Newton–Raphson method with $t_{0} = 25$ to find this value of $t$ to 2 decimal places.
[3 marks]
Answer
(c)
The equation $A = B$ has a second root at $t = 0.186$ . Describe what happens to the two populations between $t = 0$ and $t = 30$ .
[2 marks]
(d)
Give one reason why model A is not realistic for large $t$ .
[2 marks]
Total for D5: 9 marks
D6
A company makes $x$ hundred items a week. Its weekly profit, $£ P$ thousand, is modelled by $P = 40 x e^{- 0.1 x} - 2 x - 30$ .
(a)
Show that the company stops making a profit at a value of $x$ between $25$ and $26$ .
[2 marks]
(b)
Show that $\frac{d P}{d x} = 0$ can be written $x = 10 - 0.5 e^{0.1 x}$ .
[3 marks]
(c)
Use $x_{n + 1} = 10 - 0.5 e^{0.1 x_{n}}$ with $x_{0} = 7$ to find $x_{1}$ , $x_{2}$ , $x_{3}$ and $x_{4}$ to 4 decimal places.
[3 marks]
Answer
(d)
The iterates converge to $8.795$ . Find the greatest weekly profit, to the nearest $£100$ .
[2 marks]
Answer
Total for D6: 10 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A horizontal pipe has a circular cross-section of radius $40$ cm. Water flows along the bottom of the pipe and fills $30%$ of the area of the cross-section. Find the depth of the water, to the nearest millimetre. Show that your answer is correct to this accuracy.
[9 marks]
Hint 1 · What to try
The water is a segment of the circle. Let $θ$ be the angle it makes at the centre. Its area is $\frac{1}{2} r^{2} (θ - sin θ)$ .
Hint 2 · The first line
$\frac{1}{2} r^{2} (θ - sin θ) = 0.3 π r^{2}$ gives $θ - sin θ = 0.6 π$ . Use the Newton–Raphson method, starting near $2.5$ .
Hint 3 · The full method
The depth is $40 - 40 cos (\frac{θ}{2})$ . To check the depth to the nearest millimetre, work out the fraction of the area filled at depths of $27.15$ cm and $27.25$ cm.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can set up an equation f(x) = 0 from a model and find a root numerically.
I can give a root in the units of the problem, such as hours and minutes.
I can show that a time or length is correct to the accuracy asked for.
I can check that a root makes sense in context and criticise the model.

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