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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 dPdt=10002+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 0510002+tdt.

[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 0365Ptdt 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.

1252.71250 (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=1000ln2+t+c        (M1)A1

c=20000-1000ln219306.8        (M1)A1

P=1000ln2+t+19300

 

METHOD 2

20000PdP=0t10002+xdx        (M1)

P20000P=1000ln2+x0t        A1

 

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

 

P-20000=1000ln2+t-ln2        (M1)A1

P=1000ln2+t2+20000

 

[4 marks]

b.

88478838850000 (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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