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Date May 2022 Marks available 4 Reference code 22M.2.SL.TZ2.5
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Hence and Find Question number 5 Adapted from N/A

Question

A cafe makes x litres of coffee each morning. The cafe’s profit each morning, C, measured in dollars, is modelled by the following equation

C=x10k2-3100x2

where k is a positive constant.

The cafe’s manager knows that the cafe makes a profit of $426 when 20 litres of coffee are made in a morning.

The manager of the cafe wishes to serve as many customers as possible.

Find an expression for dCdx in terms of k and x.

[3]
a.

Hence find the maximum value of C in terms of k. Give your answer in the form pk3, where p is a constant.

[4]
b.

Find the value of k.

[2]
c.i.

Use the model to find how much coffee the cafe should make each morning to maximize its profit.

[1]
c.ii.

Sketch the graph of C against x, labelling the maximum point and the x-intercepts with their coordinates.

[3]
d.

Determine the maximum amount of coffee the cafe can make that will not result in a loss of money for the morning.

[2]
e.

Markscheme

attempt to expand given expression            (M1)

C=xk210-3x31000

dCdx=k210-9x21000         M1A1


Note: Award M1 for power rule correctly applied to at least one term and A1 for correct answer.

 

[3 marks]

a.

equating their dCdx to zero            (M1)

k210-9x21000=0

x2=100k29

x=10k3            (A1)

substituting their x back into given expression            (M1)

Cmax=10k30k2-300k2900

Cmax=2k39 0.222k3           A1 

 

[4 marks]

b.

substituting 20 into given expression and equating to 426           M1 

426=2010k2-3100202

k=15           A1 

 

[2 marks]

c.i.

50           A1 

 

[1 mark]

c.ii.

              A1A1A1


Note: Award A1 for graph drawn for positive x indicating an increasing and then decreasing function, A1 for maximum labelled and A1 for graph passing through the origin and 86.6, marked on the x-axis or whose coordinates are given.

 

[3 marks]

d.

setting their expression for C to zero  OR  choosing correct x-intercept on their graph of C              (M1)

xmax=86.6  86.6025 litres              A1

 

[2 marks]

e.

Examiners report

[N/A]
a.
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b.
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c.i.
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c.ii.
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d.
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e.

Syllabus sections

Topic 5—Calculus » SL 5.6—Stationary points, local max and min
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Topic 5—Calculus

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