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Trigonometry and Modelling

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Question 64

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

i.

Solve, for 0<x⩽π0 < x \leqslant \pi0<x⩽π, the equation

6sin⁡xtan⁡x+11=cos⁡x 6 \sin x \tan x + 11 = \cos x 6sinxtanx+11=cosx

giving your answer in radians to 3 significant figures.

[4]
a.

A hydrographic surveyor models the water depth in a shipping channel. The depth, HHH metres, ttt hours after midnight, is modelled by the equation

H=8.5+4.2sin⁡(kt+15)∘0⩽t<24 H = 8.5 + 4.2 \sin(kt + 15)^\circ \quad 0 \leqslant t < 24 H=8.5+4.2sin(kt+15)∘0⩽t<24

where kkk is a constant. Use the equation of the model to answer parts (a) to (c).

Given that

  • the water depth was 10.610.610.6 m at 10 am
  • 0<k<200 < k < 200<k<20

Find all possible values for kkk, giving each answer to 2 decimal places.

[4]
b.

Given further that 10<k<2010 < k < 2010<k<20

Find the maximum water depth in the shipping channel,

[1]
c.

Find the time of day at which this maximum depth occurs. Give your answer to the nearest minute.

[2]

Trigonometry and Modelling Questions

  1. A Level
  2. /Maths
  3. /Trigonometry and Modelling