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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 What 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 kmh1 and walk at 6 kmh1. Let PˆAB=θ radians, and t be the time in hours taken by Jorg to travel from A to B.

Show that t=23(2cosθ+θ).

[3]
a.

Find the value of θ for which dtdθ=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

cosθ=AP4AP=4cosθ     A1

Note: Allow correct use of cosine rule.

 

arc PB=2×2θ=4θ     A1

t=AP3+PB6     M1

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

 

t=4cosθ3+4θ6=4cosθ3+2θ3=23(2cosθ+θ)     AG

[3 marks]

a.

dtdθ=23(2sinθ+1)     A1

23(2sinθ+1)=0sinθ=12θ=π6 (or 30 degrees)     A1

[2 marks]

b.

d2tdθ2=43cosθ<0(at θ=π6)     M1

t is maximized at θ=π6     R1

time needed to walk along arc AB is 2π6 (1 hour)

time needed to row from A to B is 43 (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 6 - Core: Calculus » 6.3 » Optimization problems.

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