Matthew is riding a ferris wheel at a constant speed of 2 revolutions per minute. The boarding height for the wheel is 1m, and the radius is 7m.
a) What is the equation of the function that describes Matthew's Height in (m) in terms of time in minutes, assuming matthew starts at the lowest point of the wheel?
b) Sketch one cycle of the of function
c) What is Matthew's height after 18 minutes?
The period is π , and the amplitude is 7
so we could go with y = 7sin πt
we want the min to be 0 and the max to be 14
so we need y = 7sin πt + 7
when t = 0 we want y = 0, so we need a phase shift
y = 7sin π(t+Ø) + 7
when t=0, we want y = 0
when t = 1/2 we want y = 7
when t = 1 , we want y = 14
et
so using t=1, y = 14
14 = 7sin π(1+Ø) +7
1 = sin π(1+Ø)
I know that sin π/2 = 1
so π(1+Ø) = π/2
1+ Ø = 1/2
Ø = -1/2
a possible equation:
y = 7 sin π(t - 1/2) + 7 or y = 7 sin (πt - π/2) + 7
checking:
when t = 0 , y = 0
when t = 1/2, y = 7
when t = 1 , y = 14
looks good ....
when t = 18/60 or 3/10
y = 7 sin π(3/10 - 1/2) + 7
= appr 2.89 m
confirmation:
http://www.wolframalpha.com/input/?i=plot+y+%3D+7+sin+(%CF%80x+-+%CF%80%2F2)+%2B+7
a) To find the equation describing Matthew's height in relation to time, we can use the equation for the height of an object moving in a circle, which is given by:
h(t) = r + a * cos(b * t)
where:
- h(t) is the height of the object at time t
- r is the radius of the ferris wheel (7m in this case)
- a is the amplitude, which is the maximum height above the radius (in this case, 1m)
- b is the angular speed, which is given by 2π times the number of revolutions per minute (2 revolutions/minute in this case)
- t is the time in minutes
So, the equation for Matthew's height would be:
h(t) = 7 + 1 * cos(2π * 2 * t)
b) To sketch one cycle of the function, we need to see how the height of the wheel changes over a specific time interval. Let's consider one cycle as 0 to 1 minute:
- At t = 0 minutes, Matthew is at the lowest point of the wheel, so his height is r (7m).
- At t = 0.5 minutes, half of a cycle has passed, so he would be at the highest point of the wheel, which is r + a (7 + 1 = 8m).
- At t = 1 minute, one full cycle has passed, and he is back at the lowest point of the wheel, so his height is r (7m).
So, the sketch would show a sinusoidal function that starts at 7m, goes up to a maximum of 8m, and then comes back down to 7m.
c) To find Matthew's height after 18 minutes, we can substitute t = 18 into the equation:
h(18) = 7 + 1 * cos(2π * 2 * 18)
Simplifying, we get:
h(18) = 7 + 1 * cos(72π)
Calculating the cosine of 72π, we get approximately -0.99999999998.
So, Matthew's height after 18 minutes is approximately:
h(18) = 7 + 1 * (-0.99999999998) = 6.00000000002m