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The question is from checkpoint book 11-14 from section 4, chapter 19, Shape, Space and measures. Question number 8

A ferris wheel, centre O, has a diameter of 10m and carries eight equally spaced carriages for children to sit in. The carriages are numbered from 1-8.

I've query regarding a subquestion...

(c) the shortest distance between carriages 1 and 4.

I saw checkpoint answer sheets, answer is 9.2m.

Any help regarding above is really appreaciated.

Thanks, Arif

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Let point $P$ denote the first carriage and point $Q$ denote the second carriage. We observe the following about the triangle $OPQ$:

  • Both $OP$ and $OQ$ have length 5 (the radius of the ferris wheel).
  • The angle $POQ$ has measure $3\pi / 4$ (the carriages are equally spaced and there are eight of them).

Let $c$ denote the length of the side $PQ$. Using the law of cosines, $$ \begin{align*} c^2 &= 5^2 + 5^2 - 2\cdot 5 \cdot 5 \cdot \cos(3\pi /4)\\ &= 50 + 25\sqrt{2}, \end{align*} $$ and so $c = \sqrt{50 + 25 \sqrt{2}} \approx 9.2$.

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Hint: use the law of cosines. What is the angle between carriages 1 and 4?

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