TRIG MATH HELP!!

Question

A Ferris wheel moves with constant speed and completes one rotation every 40 seconds. The wheel has a radius of 12 m and its lowest point is 2 m above the ground. The height of a chair on the Ferris wheel above ground can be modelled by the function, h(t) = a cos bt + c, where t is the time in seconds.

a) Find the value of a, b and c.

The chair first reaches a height of 20 m

above the ground after p seconds.

b) Find the angle that the chair has rotated

through to reach this position.

c)Find the value of p.

Answer 1

a) The time period of rotation is T=40 sec

The angular frequency is given as ω =2π/T = 2π/40=π/20 rad

In the given expression for height ( a cos bt +c), b represents the angular frequency.

So b= π/20 rad

When t=0, the ferry wheel's height is equal to 2 m. i.e.

h(0)=a cos(0)+c=2

=> a+c=2..........(1)

Now given that the radius of ferry wheel is 12 m. When half of the rotation is completed, the height of the ferry wheel is 2+12+12=26 m above the ground and time passed is half of the time period i.e 20 secs. so

a cos(π/20 * 20)+c=26

=> -a+c=26 ...........(2)

solving equation 1 and 2, we get

c=14, a=-12

b) The angle of rotation is given by b*t (= b*p in this part) and let's say it's θ when h=20 m. So

20 = -12cos θ +14 => cosθ=-6/12=-1/2

=> θ=120 deg = 2π/3 rad

The time p is given as

b*p= 2π/3

=> π/20 *p= 2π/3

=> p= 40/3 sec

Answer 2

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