The height h(t) (in feet) of the seat of a child's swing above ground level is given by h(t) = -1.1cos(2pi/3t)+3.1 where t is the time in seconds after the swing is set in motion.
a) Find the maximum and minimum height of the swing.
you know that the max of cos(x) is 1 and the min is -1
so, A cosx oscillates between A and -A
So, this function oscillates between
3.1+1.1 and 3.1-1.1
or
4.2 and 2.0
To find the maximum and minimum height of the swing, we need to find the maximum and minimum values of the function h(t) = -1.1cos(2π/3t) + 3.1.
The maximum and minimum values of a cosine function can be found by considering the amplitude and vertical shift of the function.
In this case, the amplitude is 1.1, which represents the distance between the highest and lowest points on the swing (half the difference between the maximum and minimum heights).
The vertical shift is 3.1, which represents the average height of the swing.
Therefore, the maximum height can be found by adding the amplitude (1.1) to the vertical shift (3.1), and the minimum height can be found by subtracting the amplitude (1.1) from the vertical shift (3.1).
Maximum height = 3.1 + 1.1 = 4.2 feet
Minimum height = 3.1 - 1.1 = 2.0 feet
So, the maximum height of the swing is 4.2 feet and the minimum height is 2.0 feet.
To find the maximum and minimum height of the swing, we need to analyze the given equation h(t) = -1.1cos(2π/3t) + 3.1.
The maximum and minimum values of a cosine function occur when the cosine function reaches its maximum and minimum values, which are 1 and -1, respectively.
In the given equation, the cosine function is multiplied by -1.1 and then added to 3.1. This means that the maximum value is obtained when the cosine function is -1 and the minimum value is obtained when the cosine function is 1.
To find the maximum and minimum heights, substitute 1 for cos(2π/3t) in the equation and solve for h(t):
h(t) = -1.1cos(2π/3t) + 3.1
h(t) = -1.1(1) + 3.1
h(t) = -1.1 + 3.1
h(t) = 2
Therefore, the maximum height of the swing is 2 feet.
Next, substitute -1 for cos(2π/3t) in the equation and solve for h(t):
h(t) = -1.1cos(2π/3t) + 3.1
h(t) = -1.1(-1) + 3.1
h(t) = 1.1 + 3.1
h(t) = 4.2
Therefore, the minimum height of the swing is 4.2 feet.
In summary, the maximum height of the swing is 2 feet and the minimum height is 4.2 feet.