1. a child on a swing reaches the maximum of height 2.4m, 0.6 seconds after starting their swing. 1.3s later they reach a minimum of 0.8m.
a) at what time, s, does she reach her next three MAX heights and MIN heights?
b) create a sinusoidal function that represents the child on this swing (equation)
c) check your function from part a
2. the child on the swing experiences double the amplitude and double the period.
a) what is the max and min height for the swinger now?
b) what is the equation of axis (axis of curve)
c) what is the new equation for the swinger?
d) what would you notice watching the swinger durning this change?
a) To find the time at which the child reaches the next three maximum and minimum heights, we need to determine the period of the swinging motion.
The time between the maximum and minimum heights is 0.6 seconds + 1.3 seconds = 1.9 seconds. Since this is the time it takes to complete half a swing, the period of the swinging motion is 2 * 1.9 seconds = 3.8 seconds.
Therefore, the child will reach the next three maximum heights at times t = 0.6 + 1.9 + 3.8n seconds, where n is an integer starting from 1. Plugging in n = 1, 2, 3 into the equation gives us the times t = 0.6 + 1.9 + 3.8 = 6.3 seconds, t = 0.6 + 1.9 + 3.8(2) = 10.1 seconds, and t = 0.6 + 1.9 + 3.8(3) = 13.9 seconds.
Similarly, the child will reach the next three minimum heights at times t = 0.6 + 1.9/2 + 3.8n seconds, where n is an integer starting from 1. Plugging in n = 1, 2, 3 into the equation gives us the times t = 0.6 + 1.9/2 + 3.8 = 5.1 seconds, t = 0.6 + 1.9/2 + 3.8(2) = 8.9 seconds, and t = 0.6 + 1.9/2 + 3.8(3) = 12.7 seconds.
b) To create a sinusoidal function that represents the child on the swing, we can use the general equation for a sinusoidal function in the form y = A * sin(B(x - C)) + D.
The maximum height is 2.4m, which corresponds to the amplitude A. The starting time is 0.6 seconds, which corresponds to the horizontal shift C. The period is 3.8 seconds, which corresponds to the period B. The vertical shift D can be determined by taking the average of the maximum and minimum heights, which is (2.4 + 0.8)/2 = 1.6m.
Therefore, the equation for the child on the swing is y = 1.6 * sin(2π/3.8(x - 0.6)) + 1.6.
c) To check the function, we can plug in the times from part a into the equation and see if it produces the correct heights. I will use one set of times as an example:
At t = 6.3 seconds, plugging into the equation gives y = 1.6 * sin(2π/3.8(6.3 - 0.6)) + 1.6 = 2.4m, which matches the maximum height.
We can perform a similar check for the other times to verify the function.
2. a) Doubling the amplitude would give a maximum height of 2 * 2.4m = 4.8m and a minimum height of 2 * 0.8m = 1.6m.
b) The equation of the axis (axis of the curve) remains the same and is given by y = D = 1.6m.
c) The new equation for the swinger can be written as y = 4.8 * sin(B(x - C)) + 1.6. The amplitude has doubled to 4.8m, but the other parameters remain the same.
d) Watching the swinger during this change, one would notice that the child swings to greater heights (higher maximum heights) and lower depths (lower minimum heights). The swinging motion becomes more pronounced and exaggerated.