A ferris wheel has a radius of 9 m, making a full rotation every 16 seconds. the bottom of the wheel is 1.5 m off the ground. what is the equation (sin or cos) that models the height of a chair on the ferris wheel that starts at the top of the ride. where is x in seconds?
Remember to use the equation y=a*cos(bx+c)+d where:
|a| is the amplitude
2π/|b| is the period, or how long the Ferris wheel takes to get back to its starting point
-c/b is the phase shift/horizontal displacement
d is the vertical shift/midline
It's better to use cosine rather than sine because the cosine function always starts at the maximum, which makes sense in the context of this problem given that the chair starts at the top of the ride.
The radius is 9m, so this means that our amplitude is |a|=|9|=9
It takes the Ferris wheel 16 seconds to make one full rotation and come back to its starting point, so our period is equal to 2π/|b|=16 where b=π/8
There's no indication of any horizontal movement, so there's no phase shift
Since the bottom of the wheel is 1.5m off the ground, this means our midline is d=1.5
Plugging in all our variables, we have for our equation y=9cos(π/8*x)+1.5 where the time elapsed is x>0
Midline should be d=10.5 not d=1.5 because it's 1.5 meters off the ground. Correct equation is y=9cos(π/8*x)+10.5
ohh i get it now, tysm!
No problem! Let me know if you have any more questions!
radius of 9 m
y = 9sin(x)
full rotation every 16 seconds
y = 9sin(π/8 x)
the bottom of the wheel is 1.5 m off the ground.
y = 10.5 + 9sin(π/8 x)
starts at the top of the ride
That means we want
y = 10.5 + 9cos(π/8 x)
To model the height of a chair on the ferris wheel as a function of time, we can use the cosine function. The cosine function is ideal for this scenario because it describes the vertical position of an object moving in a circular path.
Let's break down the problem and derive the equation step by step:
1. The radius of the ferris wheel is given as 9 m, which means it completes a full rotation every 2π radians (or 360 degrees).
- One full rotation corresponds to a complete period of the function.
2. The ferris wheel takes 16 seconds to complete one full rotation.
- The period of the function is equal to the time it takes for one full rotation, which is 16 seconds.
3. We are given that the bottom of the wheel is 1.5 m off the ground.
- This means that at the lowest point of the ferris wheel's motion, the height is 1.5 m.
4. The chair on the ferris wheel starts at the top of the ride.
- This implies that at the beginning, when time (x) is 0 seconds, the chair is at its highest point.
Based on these observations, the equation for the height (h) of the chair on the ferris wheel as a function of time (x) can be written as:
h(x) = A * cos(Bx + C) + D
where:
- A is the amplitude of the function, equal to the radius of the ferris wheel (9 m).
- B is the angular frequency, calculated as 2π divided by the period of the function (2π/16), which simplifies to π/8.
- C is the phase shift, determined by the starting point of the chair. Since the chair starts at the top of the ride, the phase shift is 0.
- D is a vertical shift, representing the offset from the ground level. In this case, it is -1.5 m since the bottom of the wheel is 1.5 m off the ground.
Substituting the given values, the equation becomes:
h(x) = 9 * cos((π/8)x + 0) - 1.5
Therefore, the equation that models the height of the chair on the ferris wheel is:
h(x) = 9 * cos((π/8)x) - 1.5
In this equation, x represents the time in seconds, and h(x) represents the height of the chair in meters at that specific time.