the ferris wheel at an amusement park measures 16m in diameter. the wheel does 3 rotations every minute. the bottom of the wheel is 1m above the ground...
a) determine the simplest equation that models Megan's height above that ground(h) over time (t). give 2 more equations that model the situation.
first two bits of information
max is 8, min is -8, so a = 8
period = 2?/k = 20 min
20k = 2?
k = ?/10
so let's start with h = 8 sin (?/10)t
so we have the "shape" of our curve
(sketch this from 0 to 20)
http://www.wolframalpha.com/input/?i=h+%3D+8+sin+((%CF%80%2F10)t)
this has a min of -8, but we want our minimum to be 1, so let's add 9
h = 8 sin(?/10)t + 9
http://www.wolframalpha.com/input/?i=h+%3D+8+sin+((%CF%80%2F10)t),h+%3D+8+sin((%CF%80%2F10)t)+%2B+9
notice that this has a min of 1 at t = -5, but we want that min to be at t = 0
so let's "move" our curve 5 units to the right
h = 8 sin( (?/10)(t-5) ) + 9
We could just as well used a cosine curve to do the above. As a matter of fact, Wolfram switched my last equation to a cosine curve
http://www.wolframalpha.com/input/?i=h+%3D+8+sin+((%CF%80%2F10)t),h+%3D+8+sin((%CF%80%2F10)t)+%2B+9,+h+%3D+8+sin(+(%CF%80%2F10)(t-5)+)+%2B+9
I said: period = 2π/k = 20 min
That should have been 20 seconds
To determine the simplest equation that models Megan's height above the ground (h) over time (t), we need to consider the relationship between the height, time, and the characteristics of the ferris wheel.
Let's break down the key information given:
1. The diameter of the ferris wheel is 16m. The radius (r) can be found by dividing the diameter by 2: r = 16m/2 = 8m.
2. The Ferris wheel completes 3 rotations every minute. This means it takes 1/3 of a minute (or 20 seconds) for one full rotation.
3. The bottom of the wheel is 1m above the ground.
Now, let's approach the problem and derive the equations:
Equation 1: Cosine Function Equation
Since the ferris wheel's motion follows a circular path, we can use the cosine function to model Megan's height. The cosine function oscillates between the values of -1 to 1, with a maximum at the initial height.
The equation is: h(t) = A * cos(B(t - C)) + D
In this equation,
- A represents the amplitude of the cosine function, which is the maximum height above the ground (h). In this case, it is the radius of the wheel (8m).
- B represents the frequency of the oscillation. Since the ferris wheel completes one full rotation in 20 seconds, the frequency is 2π/20 radians per second.
- C represents the horizontal shift or phase shift, which determines the starting point of the cosine function. Since the starting point mentioned is not specified, we can assume it to be 0 or leave a placeholder.
- D represents the vertical shift or the height offset from the ground. In this case, it is 1m.
So, the equation becomes: h(t) = 8 * cos((2π/20)t) + 1
Equation 2: Sine Function Equation
Alternatively, the sine function can also be used to model the height of the ferris wheel. The equation is similar to the cosine function but with a phase shift of π/2.
The equation is: h(t) = A * sin(B(t - C)) + D
In this equation, the values of A, B, C, and D remain the same as in equation 1, except for C which becomes π/2.
So, the equation becomes: h(t) = 8 * sin((2π/20)t + π/2) + 1
Equation 3: Linear Equation
Another equation that can model the situation is a linear equation that represents a constant rate of change in height.
Since the ferris wheel does 3 rotations every minute, one complete rotation corresponds to 20 seconds (1/3 of a minute). Therefore, we can determine the height above ground by dividing the time elapsed by the time for one rotation and multiplying by the full height of the wheel (16m).
The equation is: h(t) = (16/20)t + 1
In this equation, (16/20) represents the slope or rate of change, t represents the time in seconds, and 1 represents the height offset from the ground.
These are the three equations that can model Megan's height above the ground over time.