Question

The largest Ferris Wheel in the world is the London Eye in England. The height (in meters) of a rider on the London Eye after t minutes can be described by the function 

https://gyazo.com/ae58de99e85b409da342e3a5ee985a0b 

Answer 1

Solution:-

Consider the function representing the height of a rider on the London eye in meters after t minutes

h(t) = 65sin[12(t-7.5) + 70 ....(1)

We know that the standard equationof sin function is

y = Asin[B(x - C)] + D ....(2)

Here,

A = amplitude (radius of wheel)

B = 2π/period

C = phase shift

D = Vertical shift or average height

On comparing equations (1) and (2) we get

A = Amplitude or Radius= 65 meters ....(3)

B = 12 ....(4)

C= phase shift = 7.5 minutes....(5)

D = vertical shift = average height= 70 meters ....(6)

Now, let us use the above information to answer the given problems

(a)

Diameter

= 2 times of amplitude or radius

= 2×65

=130 meters

Hence, diameter of ferris wheel is 130 meters

(b)

At t = 0, putting t = 0 in equation (1)

h(0) = 65sin[12(0 - 7.5)] + 70

Or h(0)= 65sin(-90) + 70

Or h(0) = 65(-0.893997) + 70

Or h(0) = -58.12 +70

Or h(0) = 11.88 meters

Hence, at t = 0 , the rider is 5 meters above the ground. The significance of this value is that this is height of rider at which ferris wheel start revolving.

(c)

At the top of the wheel , the rider's height

=Average height + radius of the wheel

= 70 meters+ 65 meters

= 135 meters

Hence,at the top of the wheel , the rider is 135 metrrs high.

(d)

For height h(t) = 100 meters

Putting h(t) = 100 m in equation (1)

100 = 65sin[12(t-7.5)] + 70

Or 65sin[12(t -7.5)] = 100 -70

Or 65sin[12(t -7.5) = 30

Or sin[12(t-7.5)]= 30/65 = 0.461538

Or 12(t -7.5)= sin^​​​-1(0.461538)

Or 12(t -7.5)= 0.4797286, 2.661864

Or t -7.5 = 0.4797286/12, 2.661864/12

Or t -7.5 = 0.039977 , 0.221822

Or t = 7.5+0.039977 , 7.5+0.221822

Or t = 7.54 minutes , 7.7218 minutes

Hence at t = 7.54 minutes and t = 7.7218 minutes the height of the rider is 100 meters.

(e)

Since B= 12 (as discussed above)

So, period of ferris wheel = 2π/B = 2π/12 = π/6 or 0.5236 minutes

Hence, period of ferris wheel is π/6 or 0.5236 minutes.

(f)

Minimum bvalue of function h(x)

= Average value - amplitude

= 70 meters - 65 meters

= 5 meters

Hence, the minimum value of h(x) is 5 meters.

It represents the bottom most position of rider.