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Trigonometry and Modelling

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Question 47

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+5=2cos⁡x 6 \sin x \tan x + 5 = 2 \cos x 6sinxtanx+5=2cosx

giving your answer in radians to 3 significant figures.

[4]
a.

The dissolved oxygen concentration, DDD mg/L, in a fish pond ttt hours after midnight is modelled by the equation

D=15+10sin⁡(kt−35)∘0⩽t<24 D = 15 + 10 \sin(kt - 35)^\circ \quad 0 \leqslant t < 24 D=15+10sin(kt−35)∘0⩽t<24

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

Given that

  • the dissolved oxygen concentration was 202020 mg/L at 8 am
  • 0<k<300 < k < 300<k<30

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

[4]
b.

Given further that 0<k<150 < k < 150<k<15

Find the maximum dissolved oxygen concentration in the pond.

[1]
c.

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

[2]

Trigonometry and Modelling Questions

  1. A Level
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