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Date	November Example question	Marks available	2	Reference code	EXN.1.AHL.TZ0.10
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 0
Command term	State and Determine	Question number	10	Adapted from	N/A

Question

The production of oil $(P)$ , in barrels per day, from an oil field satisfies the differential equation $\frac{d P}{d t} = \frac{1000}{2 + t}$ where $t$ is measured in days from the start of production.

The production of oil at $t = 0$ is $20, 000$ barrels per day.

Find $\int_{0}^{5} \frac{1000}{2 + t} d t$ .

[1]

a.i.

State in context what this value represents.

[1]

a.ii.

Find an expression for $P$ in terms of $t$ .

[4]

b.

Determine $\int_{0}^{365} P (t) d t$ and state what it represents.

[2]

c.

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$1252.7 \dots \approx 1250$ (barrels per day) A1

[1 mark]

a.i.

This is the increase (change) in $P$ (production per day) between $t = 0$ and $t = 5$ (or during the first $5$ days) A1

[1 mark]

a.ii.

METHOD 1

$P = 1000 \ln (2 + t) + c$ (M1)A1

$c = 20000 - 1000 \ln 2 \approx 19306.8 \dots$ (M1)A1

$P = 1000 \ln (2 + t) + 19300$

METHOD 2

$\int_{20000}^{P} d P = \int_{0}^{t} \frac{1000}{2 + x} d x$ (M1)

${[P]}_{20000}^{P} = 1000 {[\ln (2 + x)]}_{0}^{t}$ A1

Note: A1 is for the correct integral, with the correct limits.

$P - 20000 = 1000 (\ln (2 + t) - \ln 2)$ (M1)A1

$P = 1000 \ln (\frac{2 + t}{2}) + 20000$

[4 marks]

b.

$8847883 \approx 8850000$ (barrels) A1

Total production of oil in barrels in the first year (or first $365$ days) A1

Note: For the final A1 “barrels”’ must be present either in the statement or as the units.

Accept any value which rounds correctly to $8850000$

[2 marks]

c.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

[N/A]

c.

Syllabus sections

Topic 5—Calculus » SL 5.5—Integration introduction, areas between curve and x axis

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