a ferris wheel with a 40-ft diameter rotates once every 30 seconds. the bottom of the wheel is located 1.5 feet above the ground. you get on at the very bottom of the ferris wheel at time t = 0 and then the ferris wheel begins to turn counter-clockwise. for the problems below, assume the wheel makes 2 full rotations.
a. at what times are you at the top of the ferris wheel? what is the height at the top of the ferris wheel?
b. at what times are you at the bottom of the ferris wheel? what is the height at the bottom of the ferris wheel?
c. at what times are you in the exact middle of the ferris wheel? what is the height of the exact middle of the ferris wheel?
d. write a cosine function to model your height above the ground over time, t.
a. 15, 45, 75, and 105 Seconds.
The elapsed time between each of the 4 numbers is 30 seconds which is
the time for one rev. 15s = 1/2 rev.
h = 1.5 + 40 = 41.5 Ft. above gnd.
b. 30, 60, 90, and 120 Seconds.
h = 1.5Ft. above gnd.
c. (3/4)rev * 30s/rev = 22.5 s.
22.5, 52.5, 82.5, and 112.5 Seconds.
h = 1.5 + 0.5*40 = 21.5 Ft.
a. To find the times when you are at the top of the Ferris wheel, we need to determine how long it takes for the wheel to make a full rotation. Since it takes 30 seconds for the Ferris wheel to complete one rotation, it will take twice as long, or 60 seconds, for it to make two full rotations.
Using this information, we can find the times when you are at the top by dividing the total time interval (60 seconds) into four equal parts. This is because there are four positions of interest - top, bottom, middle, and bottom again.
So, the times when you are at the top of the Ferris wheel are:
1st time: t = 15 seconds
2nd time: t = 30 seconds
3rd time: t = 45 seconds
4th time: t = 60 seconds
To find the height at the top of the Ferris wheel, we need to find the maximum height. The Ferris wheel has a 40-ft diameter, so its radius is half of that, or 20 feet. Since the bottom of the wheel is located 1.5 feet above the ground, the height at the top is the sum of the radius and the bottom height. Therefore, the height at the top is 20 + 1.5 = 21.5 feet.
b. To find the times when you are at the bottom of the Ferris wheel, we can use the same intervals as before.
The times when you are at the bottom of the Ferris wheel are:
1st time: t = 0 seconds
2nd time: t = 30 seconds (after one full rotation)
3rd time: t = 60 seconds (after two full rotations)
The height at the bottom of the Ferris wheel is simply the bottom height, which is 1.5 feet.
c. To find the times when you are in the exact middle of the Ferris wheel, we can again divide the total time interval (60 seconds) into four equal parts.
The times when you are in the exact middle of the Ferris wheel are:
1st time: t = 7.5 seconds
2nd time: t = 22.5 seconds
3rd time: t = 37.5 seconds
4th time: t = 52.5 seconds
The height in the exact middle of the Ferris wheel is equal to the bottom height, which is 1.5 feet.
d. To write a cosine function to model your height above the ground over time, t, we need to use the formula for the height of a point on the Ferris wheel as a function of time.
The general formula for the height of a point on a Ferris wheel is:
h(t) = A + B * cos(C * (t - D))
Where:
A = bottom height = 1.5 feet
B = amplitude = radius of the wheel = 20 feet
C = angular frequency = 2π / period = 2π / 60 seconds = π / 30 seconds
D = phase shift = 0 (since we start at t = 0)
Substituting the values into the formula, we get:
h(t) = 1.5 + 20 * cos(π / 30 * t)
Therefore, the cosine function to model your height above the ground over time, t, is h(t) = 1.5 + 20 * cos(π / 30 * t).
To solve this problem, we need to make use of some basic concepts related to circular motion. Let's break down the problem step by step and find the answers.
a. To find the times when you are at the top of the Ferris wheel, we need to consider that each full rotation of the wheel takes 30 seconds. Since the wheel makes 2 full rotations, it will take a total of 60 seconds (2 x 30 seconds) to complete them.
The height at the top of the Ferris wheel can be found by considering the distance between the center of the wheel and the top point. Since the diameter is 40 ft, the radius (half the diameter) is 20 ft. However, we also need to account for the initial height of 1.5 ft. Therefore, the height at the top of the Ferris wheel is 1.5 ft + 20 ft = 21.5 ft.
To summarize, you will be at the top of the Ferris wheel at time intervals of 60 seconds, and the height at the top will be 21.5 ft.
b. Similar to part a, to find the times when you are at the bottom of the Ferris wheel, we also need to consider that each full rotation of the wheel takes 30 seconds. So, it will take a total of 60 seconds to complete 2 full rotations.
The height at the bottom of the Ferris wheel is the distance between the center and the lowest point. Since the radius is 20 ft and there is an initial height of 1.5 ft, the height at the bottom of the Ferris wheel is 1.5 ft - 20 ft = -18.5 ft.
To summarize, you will be at the bottom of the Ferris wheel at time intervals of 60 seconds, and the height at the bottom will be -18.5 ft.
c. To find the times when you are in the exact middle of the Ferris wheel, we need to consider that the exact middle corresponds to the height halfway between the top and the bottom. So, the height of the exact middle is (-18.5 ft + 21.5 ft) / 2 = 1.5 ft.
Since the middle point is halfway through the rotation, it will take half the time of a full rotation to reach it. Therefore, you will be at the exact middle of the Ferris wheel at time intervals of 30 seconds.
d. To write a cosine function to model your height above the ground over time, t, we first need to determine the amplitude and period.
The amplitude is the maximum displacement from the middle position, which in this case is (21.5 - 1.5) / 2 = 10 ft.
The period is the time it takes for one complete cycle, which in this case is 60 seconds (2 x 30 seconds).
Thus, the equation for your height above the ground, h(t), will be:
h(t) = amplitude * cosine(2πt / period)
Substituting the values, we get:
h(t) = 10 * cosine(2πt / 60)