You have been riding the London Eye Ferris wheel for 17 minutes and 30 seconds. What is your height above the ground?

Supose the center of the wheel in coordinates oringin.

This wheel rotates 360º every 30 min. So in 17,5 min will rotate 210º because: the direct proportionality of rotation in a time interval

#360/x=30/17.5#

Starting rotation at the lower position, the cabin will be at 60º to be in horizontal postion and at 200 ft to ground. Thus, applying trigonometry

#sin 60º=sqrt3/2=h/200# and from this #h=100sqrt3#

#h'=173.205 ft+ 200 ft=373.205 ft#

Given that the time period of rotation of LW is
#T=30"min"=1800s #
So angular frequecny of rotation #omega=(2pi)/T=(2pi)/1800# rad/s.

The diameter of the wheel is #400ft#

Let the mimimum height of the wheel seat at the moment to hop on for boarding it at #t=0# is # zero# and after 30 minute of riding it returns at the same lowest position. In the mean time after 15 minute of riding the rider reaches at top position I.e. at highest position 400ft from lowest position. So the height #h(t)# is a periodic function of time #(t)# of amplitude of #200ft# and is given by the following equation

#h(t)=200-200*cos((2pit)/1800)#

For #t=17min 30s=17.5" min"=17.5*60s#

#h(17.5xx60)=200-200*cos((2pi)/1800*17.5*60)#

#=200-200*cos((7pi)/6)#

#=200-200*cos(pi+pi/6)#

#=200+200*cos(pi/6)#

#=200+200*sqrt3/2#

#=200+173.205=373.205#ft