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The blades of a wind turbine have a diameter of 16m and rotate clockwise at a constant speed, 1 revolution every 4 seconds. The blades are fixed on a shaft such that the tips of the blades are always at least 7m above the ground. The point Q lies at the tip of one of the blades.

Let h be the height, in meters, of Q above the ground. After t minutes, h is given by h(t)=acos(bt)+c, where a,b,cR and a>0.

  1. Show that Q starts at the highest possible point.

    [2]
  2. Find the values of a, b and c.

    [4]

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AI
Let's analyze the wind turbine problem. First, we need to show that point Q starts at its highest position. What do you notice about the cosine function at t=0?
Well, when t=0, cos(0) equals 1, which is the maximum value of cosine. So h(0)=a+c would be the highest point?
U
AI
Exactly! Since a is positive, acos(bt) reaches its maximum when cos(bt)=1, which happens at t=0. This proves Q starts at its highest point.

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Solution to (a)

To show that point Q starts at the highest possible point, we need to analyze the function h(t)=acos(bt)+c at t=0.

Evaluate the Function at t=0

Substitute t=0 into the function :

h(0)=acos(0)+c=a1+c=a+c

At t=0, the height of point Q is a+c.


Determine the Maximum Value of the Cosine Function

The cosine function, cos(θ), has a range of [−1,1]. Therefore, the maximum value of cos(θ) is 1.

Find the Maximum Height

Since a>0, the term acos(bt) achieves its maximum value when cos(bt)=1. Thus, the maximum value of h(t) is:

hmax=a1+c=a+c


Conclusion

At t=0, the height h(0)=a+c is indeed the maximum height that h(t) can achieve, confirming that point Q starts at the highest possible point.

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