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Date	May 2022	Marks available	1	Reference code	22M.2.SL.TZ2.5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	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

$C = \frac{x}{10} (k^{2} - \frac{3}{100} x^{2})$

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 $\frac{d C}{d x}$ 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 $p k^{3}$ , 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)

$C = \frac{x k^{2}}{10} - \frac{3 x^{3}}{1000}$

$\frac{d C}{d x} = \frac{k^{2}}{10} - \frac{9 x^{2}}{1000}$ M1A1

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

[3 marks]

a.

equating their $\frac{d C}{d x}$ to zero (M1)

$\frac{k^{2}}{10} - \frac{9 x^{2}}{1000} = 0$

$x^{2} = \frac{100 k^{2}}{9}$

$x = \frac{10 k}{3}$ (A1)

substituting their $x$ back into given expression (M1)

$C_{max} = \frac{10 k}{30} (k^{2} - \frac{300 k^{2}}{900})$

$C_{max} = \frac{2 k^{3}}{9} (0.222 \dots k^{3})$ A1

[4 marks]

b.

substituting $20$ into given expression and equating to $426$ M1

$426 = \frac{20}{10} (k^{2} - \frac{3}{100} {(20)}^{2})$

$k = 15$ 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)

$x_{max} = 86.6 (86.6025 \dots)$ litres A1

[2 marks]

e.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.

[N/A]

e.

Syllabus sections

Topic 5—Calculus » SL 5.7—Optimisation

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18M.1.SL.TZ2.S_9a:
Express h in terms of r.
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Find the the rate of change of $f$ at B.
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Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
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Find the surface area to be covered by the sealant, given that the bird bath has maximum volume.
EXN.2.SL.TZ0.2e:
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21N.1.AHL.TZ0.15a:
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EXN.2.SL.TZ0.2a:
Find an expression for the total length of the ribbon $L$ in terms of $x$ and $y$ .
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EXN.2.SL.TZ0.2d:
Solve $\frac{d L}{d x} = 0$
21N.1.SL.TZ0.12a:
Find $S' (x)$ .
21N.1.SL.TZ0.12b.ii:
Interpret your answer to (b)(i) in context.

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Topic 5—Calculus