Find a formula and graph the function:
A ferris wheel is 20 meters in diameter and boarded in the six oclock position from a platform that is 4 meters above the ground. The wheel completes one full revolution every 2 minutes. At t=0 you are in the twelve oclock position.
To find a formula and graph the function for the ferris wheel's height above the ground as a function of time, we can start by analyzing the characteristics given in the problem.
We know that the ferris wheel is 20 meters in diameter. Since the wheel completes one full revolution every 2 minutes, we can assume that it travels at a constant speed. Therefore, it covers the circumference (C) of the wheel in 2 minutes.
The formula for the circumference is C = πd, where d is the diameter. Substituting the given diameter of 20 meters, we have C = π * 20 = 20π meters.
Since the wheel takes 2 minutes to complete one full revolution, we can find its angular velocity (ω) using the formula ω = 2π / T, where T is the time period for one revolution. Substituting T = 2 minutes, we get ω = 2π / 2 = π radians per minute.
To find the height above the ground (h) at a given time t, we can use the equation of a sinusoidal function:
h(t) = A * sin(ωt + φ) + h0
Where:
A is the amplitude of the function (half the vertical distance between the highest and lowest points),
ω is the angular velocity,
t is the time,
φ is the phase shift (horizontal shift of the graph), and
h0 is the vertical shift (the height above the ground at t = 0).
In our case, the amplitude can be determined by half of the diameter, A = 20 / 2 = 10 meters. The angular velocity is π radians per minute, and the phase shift is 0, since we start at the twelve o'clock position. The vertical shift is 4 meters, which corresponds to the height above the ground when t = 0.
Therefore, the formula for the height above the ground as a function of time is:
h(t) = 10 * sin(πt) + 4
To graph this function, you can plot the height above the ground (h) on the y-axis and the time (t) on the x-axis. Use the formula h(t) = 10 * sin(πt) + 4 to calculate the heights at different time intervals, and then plot the points on a graph. Connect the points to obtain a smooth curve, which represents the variation of height as the ferris wheel moves.