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. please explain! :)
I did this a few days ago
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To determine the equation that models Megan's height above the ground (h) over time (t), we need to consider the vertical motion of the Ferris wheel.
The height of an object in vertical motion can be modeled by a sinusoidal function, specifically a cosine function since the Ferris wheel starts at its highest point. The general form of a cosine function is:
h = A*cos(B(t - D)) + C
Where:
- A is the amplitude (half the height difference between the highest and lowest points)
- B affects the period (the time it takes for one complete cycle)
- D is the horizontal shift (phase shift) measured in seconds
- C represents any vertical shift (displacement)
Now, let's assign values to the variables based on the given information:
- The diameter of the Ferris wheel is 16m, so the radius (r) is half of that, which is 8m. Therefore, the amplitude (A) is 8m.
- The wheel does 3 rotations every minute, which means it completes 3 cycles in 60 seconds. So, the period (T) is 60/3 = 20 seconds. Therefore, B = 2π/T = 2π/20 = π/10.
- The bottom of the wheel is 1m above the ground, so the vertical shift (C) is 1m.
Substituting the values into the general form equation, we get:
h = 8*cos((π/10)(t - D)) + 1
This equation models Megan's height above the ground over time.
Now, let's provide two more equations that model the situation by considering different scenarios:
1. Equation to model the height above the ground from the highest point:
In this scenario, we consider the Ferris wheel initially at the highest point.
h = -8*cos((π/10)(t - D)) + 9
2. Equation to model the height above the ground from the lowest point:
In this scenario, we consider the Ferris wheel initially at the lowest point.
h = 8*cos((π/10)(t - D)) + 1
These equations will represent Megan's height at various starting positions on the Ferris wheel.