Suppose you get on a Ferris Wheel that has a diameter of 40 meters and turns at a rate 1.5 revolutions per minute (the period is not 1.5 ) the height above the ground in meters, of your seat after t minutes can be modeled by an equation. Assume you get on the ride at t=0 min and your seat is 1 meter of the ground at this time, sketch a picture of the function and find the period and find the equation.
2.How long after the ride starts will your seat be 31 meters above the ground?
3. Assume that the ride malfunctions and stops at 30 seconds. How high are you off the ground?
To sketch the function and find the equation, we need to understand the properties of a Ferris Wheel and how it relates to height and time.
1. Sketching the function:
The Ferris Wheel is a circular ride, so we can represent it as a sinusoidal function. The height of your seat above the ground will vary over time. At any point on the wheel, the height is determined by the angle relative to the ground.
To sketch the function, plot the time (t) on the x-axis and the height (h) on the y-axis. Start with t=0 and h=1. As time increases, the height will fluctuate based on the periodic nature of the Ferris Wheel.
2. Finding the period and equation:
The period of a sinusoidal function represents the time it takes for the graph to complete one full cycle before repeating. In this case, the period of the Ferris Wheel can be calculated using the formula:
Period = 1 / (rate of revolution)
Given that the rate of revolution is 1.5 revolutions per minute, we can calculate the period as follows:
Period = 1 / 1.5 = 2/3 minutes
The general equation to represent the height at any given time on the Ferris Wheel is:
h(t) = A * sin(B * (t - C)) + D
In this equation, A represents the amplitude of the function (half the difference between the maximum and minimum values), B represents the frequency (2π divided by the period), C represents a phase shift (horizontal shift of the graph), and D represents a vertical shift.
For the given problem, the amplitude is 20 meters (half of the diameter), the frequency is 2π / (2/3) = 3π radians per minute, the phase shift C = 0 (since we start at t=0), and the vertical shift D = 1 meter (starting height).
Therefore, the equation to represent the height of your seat on the Ferris Wheel is:
h(t) = 20 * sin(3πt) + 1
3. Finding the time when your seat is 31 meters above the ground:
To find the time when your seat is 31 meters above the ground, we need to solve the equation h(t) = 31. Set up the equation as follows:
31 = 20 * sin(3πt) + 1
Now, we can solve for t using algebraic methods or estimation techniques. Depending on the level of precision required, you can use numerical methods or an online graphing calculator to find the intersection point of the graph of h(t) and the line y = 31.
4. Calculating the height above the ground after 30 seconds:
To find the height when the ride stops after 30 seconds, substitute t = 30/60 = 0.5 into the equation we derived earlier:
h(0.5) = 20 * sin(3π * 0.5) + 1
Compute the expression to find the height at that time.