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Date May 2019 Marks available 1 Reference code 19M.3.AHL.TZ0.Hca_1
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Show that Question number Hca_1 Adapted from N/A

Question

A simple model to predict the population of the world is set up as follows. At time t years the population of the world is x , which can be assumed to be a continuous variable. The rate of increase of x due to births is 0.056 x and the rate of decrease of x due to deaths is 0.035 x .

Show that d x d t = 0.021 x .

[1]
a.

Find a prediction for the number of years it will take for the population of the world to double.

[6]
b.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

d x d t = 0.056 x 0.035 x        A1

d x d t = 0.021 x       AG

[1 mark]

a.

METHOD 1

d x d t = 0.021 x

attempt to separate variables       M1

1 x d x = 0.021 d t       A1

ln x = 0.021 t ( + c )       A1

EITHER

x = A e 0.021 t

2 A = A e 0.021 t       A1

Note: This A1 is independent of the following marks.

OR

t = 0 , x = x 0 c = ln x 0

ln 2 x 0 = 0.021 t + ln x 0       A1

Note: This A1 is independent of the following marks.

THEN

ln 2 = 0.021 t        (M1)

t = 33 years      A1

Note: If a candidate writes t = 33.007 , so t = 34 then award the final A1.

 

METHOD 2

d x d t = 0.021 x

attempt to separate variables      M1

A 2 A 1 x d x = 0 t 0.021 d u       A1A1

Note: Award A1 for correct integrals and A1 for correct limits seen anywhere. Do not penalize use of t in place of u .

[ ln x ] A 2 A = [ 0.021 u ] 0 t       A1

ln 2 = 0.021 t        (M1)

t = 33       A1

 

[6 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 5 —Calculus » AHL 5.18—1st order DE’s – Euler method, variables separable, integrating factor, homogeneous DE using sub y=vx
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Topic 5 —Calculus

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