13. (a) Express 10cosθ − 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90° - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 2
Question 14
13. (a) Express 10cosθ − 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90°.
Give the exact value of R and give the value of α, in degrees, to 2 decimal pl... show full transcript
Worked Solution & Example Answer:13. (a) Express 10cosθ − 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90° - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 2
Step 1
Express 10cosθ − 3sinθ in the form Rcos(θ + α)
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Answer
To express the equation in the given form, we first calculate the value of R using the formula:
R=sqrt(102+(−3)2)=sqrt109
Next, we find α using:
tan(α)=10−3
Thus, calculating α gives:
α=tan−1(10−3)≈16.70°.
Step 2
Find a complete equation for the model
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Answer
It is given that the initial height above the ground is 1 metre. Therefore, substituting the value of a:
H=1−10cos(80t)°+3sin(80t)°.
Step 3
Hence find the maximum height of the passenger above the ground
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Answer
The maximum height can be found when the cosine and sine functions reach their respective maxima.
This occurs at:
Hmax=1−10(1)+3(1)=1−10+3=−6+3=−3,
Thus the maximum height is:
Hmax=1+R=1+109=21.44 m.
Step 4
Find the time for the passenger to reach the maximum height on the second cycle
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Answer
From the equation set:
80t+16.70=540
Rearranging yields:
t=80540−16.70=6.54 minutes.
Converting this into seconds, we get:
t=6 minutes 32 seconds.
Step 5
How would you adapt the equation to reflect the increase in speed?
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Answer
To increase the speed of the Ferris wheel, the value inside the cosine and sine functions (currently at 80) needs to be increased:
H=a−10cos(kt)+3sin(kt),
where k is the new angular speed greater than 80.
