Express the height as a function of time

Question

Suppose that after you are loaded into a Ferris wheel car, the wheel begins turning at 3 revolutions per second.  The wheel has diameter 14 meters and the bottom seat of the wheel is 1 meter above the ground.  Express the height h of the seat above the ground as a function of time t seconds after it begins turning.

 

My teacher says a way to solve this is to find a sin or cos wave pattern?

Answer 1

1. let h(t) be in the A*sin(bt + c) + M form, where A = amplitude, b = 2π/k, k = 1/f, f = frequency, t = time (in seconds), c = the horizontal translation, and M = the midline.

2. A = the diameter/2 = 7

3. b = 2π/k, k = 1/3 (because the frequency is 3 rev/sec), so b = 2π/(1/3) = 6π, which makes perfect sense since one rev is 2π and 2π*3 = 6π

4. c = -π/2 because we need to translate the graph over to the right so that the function equals 1 at t = 0

5. M = 15 (the height from the ground at the top of the wheel) + 1 (height from the ground at the bottom of the wheel) divided by 2, so (15+1)/2 = 8

6. now, let's put it all together :)

7. h(t) = 7*sin(6πt - π/2) + 8

 

Let's test our equation to see how well it models the situation here:

1. h(0) should equal 1 because this is the initial height at t = 0

2. h(0) = 7*sin(6π*0 - π/2) + 8 = h(t) = 7*sin(0 - π/2) + 8 = h(t) = 7*sin(- π/2) + 8 = 7*(-1) + 8 = 1

3. that works!

4. also, h(1/6) should equal the maximum height, 15 m, because it takes 1/3 of a second for 1 rev and (1/3)*(1/2) = 1/6 for half of a revolution

5. h(1/6) = 7*sin(6π*1/6 - π/2) + 8 = 7*sin(π - π/2) + 8 = 7*sin(π/2) + 8 = 7*(1) + 8 = 15

6. that works too!

7. hence, we can confidently say: 

 

h(t) = 7*sin(6πt - π/2) + 8