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Date	November 2021	Marks available	3	Reference code	21N.2.SL.TZ0.7
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	7	Adapted from	N/A

Question

Points $A$ and $P$ lie on opposite banks of a river, such that $AP$ is the shortest distance across the river. Point $B$ represents the centre of a city which is located on the riverbank. $PB = 215 km$ , $AP = 65 km$ and $A \hat{P} B = 90 °$ .

The following diagram shows this information.

A boat travels at an average speed of $42 {km h}^{- 1}$ . A bus travels along the straight road between $P$ and $B$ at an average speed of $84 {km h}^{- 1}$ .

Find the travel time, in hours, from $A$ to $B$ given that

There is a point $D$ , which lies on the road from $P$ to $B$ , such that $BD = x km$ . The boat travels from $A$ to $D$ , and the bus travels from $D$ to $B$ .

An excursion involves renting the boat and the bus. The cost to rent the boat is $$ 200$ per hour, and the cost to rent the bus is $$ 150$ per hour.

the boat is taken from $A$ to $P$ , and the bus from $P$ to $B$ .

[2]

a.i.

the boat travels directly to $B$ .

[2]

a.ii.

Find an expression, in terms of $x$ for the travel time $T$ , from $A$ to $B$ , passing through $D$ .

[3]

b.i.

Find the value of $x$ so that $T$ is a minimum.

[2]

b.ii.

Write down the minimum value of $T$ .

[1]

b.iii.

Find the new value of $x$ so that the total cost $C$ to travel from $A$ to $B$ via $D$ is a minimum.

[3]

c.i.

Write down the minimum total cost for this journey.

[1]

c.ii.

Markscheme

$\frac{AP}{42}$ OR $\frac{215}{84}$ OR $\frac{65}{42} + \frac{215}{84}$ (M1)

time $= 4.10714 \dots$ (hours)

time $= 4.11$ (hours) A1

[2 marks]

a.i.

$AB = \sqrt{215^{2} + 65^{2}} (= 224.610 \dots)$ (A1)

time $= 5.34787 \dots$ (hours)

time $= 5.35$ (hours) A1

[2 marks]

a.ii.

$AD = \sqrt{{(215 - x)}^{2} + 65^{2}}$ (A1)

$t = \frac{\sqrt{{(215 - x)}^{2} + 65^{2}}}{42}$ (A1)

$T = \frac{\sqrt{{(215 - x)}^{2} + 65^{2}}}{42} + \frac{x}{84} (= \frac{\sqrt{x^{2} - 430 x + 50450}}{42} + \frac{x}{84})$ A1

[3 marks]

b.i.

valid approach to find the minimum for $T$ (may be seen in (iii)) (M1)

graph of $T$ OR $T' = 0$ OR graph of $T'$

$x = 177.472 \dots km$

$x = 177 km$ A1

[2 marks]

b.ii.

$T = 3.89980 \dots$

$T = 3.90$ (hours) A1

Note: Only allow FT in (b)(ii) and (iii) for $0 < x < 215$ and a function $T$ that has a minimum in that interval.

[1 mark]

b.iii.

$C = 200 \cdot \frac{\sqrt{{(215 - x)}^{2} + 65^{2}}}{42} + 150 \cdot \frac{x}{84}$ (A1)

valid approach to find the minimum for $C (x)$ (may be seen in (ii)) (M1)

graph of $C$ OR $C' = 0$ OR graph of $C'$

$x = 188.706 \dots km$

$x = 189 km$ A1

Note: Only allow FT from (b) if the function $T$ has a minimum in $0 < x < 215$ .

[3 marks]

c.i.

$C = 670.864$

$C = $ 671$ A1

Note: Only allow FT from (c)(i) if the function $C$ has a minimum in $0 < x < 215$ .

[1 mark]

c.ii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

[N/A]

c.i.

[N/A]

c.ii.

Syllabus sections

Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology

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EXN.2.SL.TZ0.6a:
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EXN.2.SL.TZ0.9d:
Sketch the graph of $T$ versus $t$ , clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.
21N.2.SL.TZ0.5b:
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Write down the minimum total cost for this journey.
21N.2.SL.TZ0.8a:
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21N.2.SL.TZ0.8c:
Find the value of $d$ .
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Find the smallest possible value of $c$ .
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Sketch the curve $y = f (x)$ , clearly indicating the coordinates of the endpoints.
22M.2.AHL.TZ2.10a.i:
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22M.3.AHL.TZ2.1c:
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22M.1.SL.TZ2.8a:
Find the coordinates of $B$ .
22M.2.SL.TZ2.8a.i:
Plant $B$ .
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21N.2.AHL.TZ0.9a:
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Find the value of $a$ .
21N.2.AHL.TZ0.9d:
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Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.
19M.2.SL.TZ2.T_6c:
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Write down the $x$ -coordinates of these two points;
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Show that $b = 0.3 - a$ .
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Find the $x$ -coordinate of $P$ .
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21M.3.AHL.TZ1.1f:
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21M.2.AHL.TZ1.12d:
Find the value of $k$ , giving your answer correct to four significant figures.
EXN.2.SL.TZ0.9f:
The model for the temperature of the water can also be expressed in the form $T = T_{0} a^{\frac{t}{10}}$ for $t \geq 0$ and $a$ is a positive constant.

Find the exact value of $a$ .
21M.2.AHL.TZ1.11f:
By considering the graph of $y = g (x) - f (x)$ , or otherwise, solve $f (x) < g (x)$ for $x \in ℝ$ .
21M.2.SL.TZ2.2b:
Find the value of $x$ for which $f' (x) = 0$ .
21N.2.AHL.TZ0.9e:
Find the height of the water at $12 : 00$ .
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Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.2.SL.TZ2.T_6b:
Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.
19M.2.SL.TZ1.T_4c:
Find the coordinates of the bridges relative to the centre of Orangeton.
20N.1.SL.TZ0.T_4a.ii:
Write down the coordinates of the local minimum point.
19M.2.SL.TZ1.T_4e:
This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
21N.2.SL.TZ0.7a.i:
the boat is taken from $A$ to $P$ , and the bus from $P$ to $B$ .
21N.2.SL.TZ0.7b.ii:
Find the value of $x$ so that $T$ is a minimum.
21N.2.SL.TZ0.7c.i:
Find the new value of $x$ so that the total cost $C$ to travel from $A$ to $B$ via $D$ is a minimum.
21N.2.SL.TZ0.8e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.7a:
Determine the value of $m$ .
21N.2.AHL.TZ0.7b:
Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .
21N.2.AHL.TZ0.9c:
Find the value of $d$ .
21N.2.AHL.TZ0.9f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9g:
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.
19M.2.SL.TZ1.T_4b.i:
Find the function, $P\left( x \right)$ , that would define this footpath on the map.
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .

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