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Date November 2011 Marks available 3 Reference code 11N.1.hl.TZ0.8
Level HL only Paper 1 Time zone TZ0
Command term Show that Question number 8 Adapted from N/A

Question

The diagram below shows a circular lake with centre O, diameter AB and radius 2 km.

 

 

Jorg needs to get from A to B as quickly as possible. He considers rowing to point P and then walking to point B. He can row at \(3{\text{ km}}\,{{\text{h}}^{ - 1}}\) and walk at \(6{\text{ km}}\,{{\text{h}}^{ - 1}}\). Let \({\rm{P\hat AB}} = \theta \) radians, and t be the time in hours taken by Jorg to travel from A to B.

Show that \(t = \frac{2}{3}(2\cos \theta + \theta )\).

[3]
a.

Find the value of \(\theta \) for which \(\frac{{{\text{d}}t}}{{{\text{d}}\theta }} = 0\).

[2]
b.

What route should Jorg take to travel from A to B in the least amount of time?

Give reasons for your answer.

[3]
c.

Markscheme

angle APB is a right angle

\( \Rightarrow \cos \theta = \frac{{{\text{AP}}}}{4} \Rightarrow {\text{AP}} = 4\cos \theta \)     A1

Note: Allow correct use of cosine rule.

 

\({\text{arc PB}} = 2 \times 2\theta = 4\theta \)     A1

\(t = \frac{{{\text{AP}}}}{3} + \frac{{{\text{PB}}}}{6}\)     M1

Note: Allow use of their AP and their PB for the M1.

 

\( \Rightarrow t = \frac{{4\cos \theta }}{3} + \frac{{4\theta }}{6} = \frac{{4\cos \theta }}{3} + \frac{{2\theta }}{3} = \frac{2}{3}(2\cos \theta + \theta )\)     AG

[3 marks]

a.

\(\frac{{{\text{d}}t}}{{{\text{d}}\theta }} = \frac{2}{3}( - 2\sin \theta + 1)\)     A1

\(\frac{2}{3}( - 2\sin \theta + 1) = 0 \Rightarrow \sin \theta = \frac{1}{2} \Rightarrow \theta = \frac{\pi }{6}\) (or 30 degrees)     A1

[2 marks]

b.

\(\frac{{{{\text{d}}^2}t}}{{{\text{d}}{\theta ^2}}} = - \frac{4}{3}\cos \theta < 0\,\,\,\,\left( {{\text{at }}\theta = \frac{\pi }{6}} \right)\)     M1

\( \Rightarrow t\) is maximized at \(\theta = \frac{\pi }{6}\)     R1

time needed to walk along arc AB is \(\frac{{2\pi }}{6}{\text{ (}} \approx {\text{1 hour)}}\)

time needed to row from A to B is \(\frac{4}{3}{\text{ (}} \approx {\text{1.33 hour)}}\)

hence, time is minimized in walking from A to B     R1

[3 marks]

c.

Examiners report

The fairly easy trigonometry challenged a large number of candidates.

a.

Part (b) was very well done.

b.

Satisfactory answers were very rarely seen for (c). Very few candidates realised that a minimum can occur at the beginning or end of an interval.

c.

Syllabus sections

Topic 3 - Core: Circular functions and trigonometry » 3.1 » The circle: radian measure of angles.

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