13. (a) Express 10cosθ - 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90º Give the exact value of R and give the value of α, in degrees, to 2 decimal places. (b) The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the equation H = a - 10cos(80º) + 3sin(80º) where a is a constant. Figure 3 shows the graph of H against t for two complete cycles of the wheel. Given that the initial height of the passenger above the ground is 1 metre, (b) (i) find a complete equation for the model, (ii) hence find the maximum height of the passenger above the ground. (c) Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle. (Solutions based entirely on graphical or numerical methods are not acceptable). (d) It is decided that, to increase profits, the speed of the wheel is to be increased. How would you adapt the equation of the model to reflect this increase in speed?

A-Level Edexcel Maths Pure Integration: 13. (a) Express 10cosθ

13. (a) Express 10cosθ - 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90º Give the exact value of R and give the value of α, in degrees, to 2 decimal places - Edexcel - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Question 15

13.-(a)-Express-10cosθ---3sinθ-in-the-form-Rcos(θ-+-α),-where-R->-0-and-0-<-α-<-90º-Give-the-exact-value-of-R-and-give-the-value-of-α,-in-degrees,-to-2-decimal-places-Edexcel-A-Level Maths Pure-Question 15-2017-Paper 2.png

13. (a) Express 10cosθ - 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90º Give the exact value of R and give the value of α, in degrees, to 2 decimal place... show full transcript

Worked Solution & Example Answer:13. (a) Express 10cosθ - 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90º Give the exact value of R and give the value of α, in degrees, to 2 decimal places - Edexcel - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Step 1

Express 10cosθ - 3sinθ in the form Rcos(θ + α)

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Answer

To express the given expression in the desired form, we start by identifying the constants involved:

Set up the equation:

$R = ext{sqrt}(10^2 + 3^2) = ext{sqrt}(100 + 9) = ext{sqrt}(109)$
To find α, we use the tangent ratio:

$an(α) = \frac{3}{10}$

To find α in degrees, calculate:

$α = an^{-1}\left(\frac{3}{10}\right) ≈ 16.70º$

Thus, the values are:

R = √109
α = 16.70º

Step 2

find a complete equation for the model

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Answer

Given the initial height of 1 metre:

Using the earlier formula:

$H = a - 10cos(80º) + 3sin(80º)$

substituting for a:

Since the height is 1 metre at t = 0:
So, we have 1 = a - 10cos(80º) + 3sin(80º)
From here:
Solve to find a = 1 + 10cos(80º) - 3sin(80º).
Thus, complete equation is:

$H = (1 + 10cos(80º) - 3sin(80º)) - 10cos(80º) + 3sin(80º)$

Step 3

hence find the maximum height of the passenger above the ground

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Answer

The maximum height typically occurs at the peak points of the trigonometric components of the equation. To find the maximum value of H, we compute:

Evaluate H when sin(80º) is maximized (at 1) and when cos(80º) is minimized (at -1):

$H_{max} = a + 10 + 3$
Substitute the value of a calculated previously:

For maximum height, calculate: $H_{max} = (1 + 10cos(80º) - 3sin(80º)) + 10 + 3$
Substitute numerical values to find H_max.

Step 4

Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle

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Answer

Using the periodic properties of H:

Set while maximizing: $80t + (16.70º) = 540$
Rearranging leads to: $t = \frac{540 - 16.70}{80}$
Compute value to find: $t ≈ 6.54$
Thus, time rounded is approximately 6 minutes and 32 seconds.

Step 5

How would you adapt the equation of the model to reflect this increase in speed?

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Answer

To adapt for speed increment:

Increase the coefficient '80' in the original equations in accordance with speed:

For example, use: $H = a - 10cos(90º) + 3sin(90º)$ 2. The increase leads to a faster cycle time, thus affecting parameters accordingly.

13. (a) Express 10cosθ - 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90º Give the exact value of R and give the value of α, in degrees, to 2 decimal places - Edexcel - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Worked Solution & Example Answer:13. (a) Express 10cosθ - 3sinθ in the form Rcos(θ + α), where R > 0 and 0 < α < 90º Give the exact value of R and give the value of α, in degrees, to 2 decimal places - Edexcel - A-Level Maths Pure - Question 15 - 2017 - Paper 2

Express 10cosθ - 3sinθ in the form Rcos(θ + α)

find a complete equation for the model

hence find the maximum height of the passenger above the ground

Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle

How would you adapt the equation of the model to reflect this increase in speed?

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