Hello, p4pri!

Did you make a sketch?

The ‘London Eye’ can be considered to be a circular frame of radius 67.5m

on the circumference of which are ‘capsules’ carrying a number of people round the circle.

Take a coordinate system where O is the base of the circle and OY is a diameter.

At any time after starting off round the frame, the capsule will be at height h metres

when it has rotated θ°.

Apparently answer is: .$\displaystyle 67.5(1-\cos\theta)$ . . . why? Code:

* * *
* *
* *
* *
* C *
- * o *
: * R * | *
: * θ | Rcosθ
R * * | *
: A o - - - + B *
: : * | *
- --+-----*-*-*------
D O

The center of the circle is $\displaystyle C.$

The capsule is at $\displaystyle A.$

$\displaystyle \angle ACO = \theta$

The radius is: .$\displaystyle CA = CO = R$

In right triangle $\displaystyle CBA\!:\;\cos\theta \:=\:\frac{CB}{R} \quad\Rightarrow\quad CB = R\cos\theta$

The height of the capsule is: .$\displaystyle h \;=\;AD \;=\;BO \;=\;CO - CB \;=\;R - R\cos\theta$

Therefore: .$\displaystyle h \;=\;R(1-\cos\theta)$