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Looking through his window, Tim has a partial view of the rotating wind turbine. The position of his window means that he cannot see any part of the wind turbine that is more than
100m above the ground. This is illustrated in the following diagram.
(f) (i) At any given instant, find the probability that point
C is visible from Tim's window.
The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.
(ii) At any given instant, find the probability that Tim can see point
C from his window. Justify your answer.
[5]

Looking through his window, Tim has a partial view of the rotating wind turbine. The position of his window means that he cannot see any part of the wind turbine that is more than $100 \mathrm{~m}$ above the ground. This is illustrated in the following diagram. $\newline$ (f) (i) At any given instant, find the probability that point $\mathrm{C}$ is visible from Tim's window. $\newline$ The wind speed increases. The blades rotate at twice the speed, but still at a constant rate. $\newline$ (ii) At any given instant, find the probability that Tim can see point $\mathrm{C}$ from his window. Justify your answer. $\newline$ [ $5$ ]

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Find probabilities using the addition rule

Full solution

Q. Looking through his window, Tim has a partial view of the rotating wind turbine. The position of his window means that he cannot see any part of the wind turbine that is more than

100 \mathrm{~m}

above the ground. This is illustrated in the following diagram.

\newline

(f) (i) At any given instant, find the probability that point

\mathrm{C}

is visible from Tim's window.

\newline

The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.

\newline

(ii) At any given instant, find the probability that Tim can see point

\mathrm{C}

from his window. Justify your answer.

\newline

[

5

]

Calculate Probability: Calculate the probability that point C is visible from Tim's window before the wind speed increases. $\newline$ Assuming the turbine is symmetrical and the blade's movement is uniform, the probability that point C is visible is the ratio of the arc that is visible from Tim's window to the total circumference of the circle described by the rotating blade. $\newline$ Let's say the visible arc is ' $v$ ' and the total circumference is ' $c$ '. The probability is then $P(C \text{ visible}) = \frac{v}{c}$ .
Determine Arc and Circumference: Determine the visible arc $v$ and the total circumference $c$ . If the turbine blade is $100$ m long, the visible arc is the part of the circle with a radius of $100$ m that is below the $100$ m height limit. Without specific measurements, we can't calculate $v$ . For the circumference, $c = 2 \cdot \pi \cdot r$ , where $r$ is the radius of the circle. Here, $r = 100$ m, so $c = 2 \cdot \pi \cdot 100$ m.
Calculate Probability: Calculate the probability $P(C \text{ visible}) = \frac{v}{c}$ with the given values. $\newline$ Since we don't have the value for $v$ , we cannot proceed with the calculation. We need more information to solve this part of the problem.

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