The table below gives the height h=f(t) in feet of a weight on a spring where t is time in seconds.
t (sec) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
h (feet) 4.8 7 8 8.3 8 7 4.8 2.6 1.6 1.3 1.6 2.6 4.8 7 8 8.3
(a) What is the period of f(t)?
1.6s
(include units)
(b) What is the the midline of f(t)?
(include units)
(c) What is the the amplitude of f(t)?
(include units)
h(0) = 4.8
h increases, goes back to 4.8 when t=12, then goes down and back to 4.8 at t=24
So what must the period be?
Hint -- it ain't 1.6s! where ever did you get that?
midline is at (max+min)/2
amplitude is (max-min)/2
Time to review the topic, right?
(a) The period of f(t) is the time it takes for the function to complete one full cycle. Looking at the table, we can see that the pattern repeats every 16 seconds (from 0s to 16s and then from 16s to 32s). Therefore, the period of f(t) is 16 seconds.
(b) The midline of f(t) is the average height of the function over one full cycle. Adding up all the given heights and dividing by the number of points, we get (4.8 + 7 + 8 + 8.3 + 8 + 7 + 4.8 + 2.6 + 1.6 + 1.3 + 1.6 + 2.6 + 4.8 + 7 + 8 + 8.3) / 16 ≈ 5.4875. Therefore, the midline of f(t) is approximately 5.4875 feet.
(c) The amplitude of f(t) is the maximum distance the function deviates from its midline. In the table, we can see that the maximum height is 8.3 feet and the minimum height is 1.3 feet. The distance between these two heights is 8.3 - 1.3 = 7 feet. Therefore, the amplitude of f(t) is 7 feet.
To find the period of f(t), we need to determine the time it takes for the function to complete one full cycle. We can do this by analyzing the given table.
(a) To find the period of f(t), we need to identify the time it takes for the function to repeat itself. Looking at the table, we can see that the values of h(t) start repeating after every 16 seconds. Therefore, the period of f(t) is 16 seconds. (include units: s)
(b) The midline of f(t) is the average of the maximum and minimum values of h(t). From the table, we can observe that the maximum value of h(t) is 8.3 feet, and the minimum value is 1.3 feet. The average of these two values is (8.3 + 1.3) / 2 = 4.8 feet. Therefore, the midline of f(t) is 4.8 feet. (include units: ft)
(c) The amplitude of f(t) is the distance between the midline and the maximum or minimum value of h(t). From the table, we can see that the maximum value of h(t) is 8.3 feet and the midline is 4.8 feet. So, the amplitude is 8.3 - 4.8 = 3.5 feet. Therefore, the amplitude of f(t) is 3.5 feet. (include units: ft)
To find the period, midline, and amplitude of the function f(t), we will analyze the given table.
(a) To find the period of f(t), we need to determine the time it takes for the function to repeat. Looking at the table, we can see that the height values seem to repeat after every 16 seconds (from t = 0 to t = 16, the height values repeat). Therefore, the period of f(t) is 16 seconds. (units: seconds)
(b) The midline of a periodic function is the horizontal line that represents its average value. For this function, we can observe that the average height is 6.1 feet. Therefore, the midline of f(t) is a horizontal line at y = 6.1 feet. (units: feet)
(c) The amplitude of a periodic function is half the distance between the maximum and minimum values of the function. Looking at the table, we can see that the maximum height is 8.3 feet and the minimum height is 1.3 feet. The amplitude is then given by (8.3 - 1.3) / 2 = 7 / 2 = 3.5 feet. Therefore, the amplitude of f(t) is 3.5 feet. (units: feet)