A man is swinging back and forth in front of Ms. Brodkey's window. At t=5s, he is at one end of the swing and 3m from the window. At t=8s he is at the other end of this swing and is 13m from window.

Question

Spider man is swinging back and forth in front of Ms. Brodkey's window. At t= 5s, he is at one end of the swing and 3 m from the window. At t= 8 s he is at the other end of this swing and is 13 m from window.Please state the period and b value for the sitation.Please state the amplitude.Please state the vertical shift.Please state the horizontal shift.Please write the equation for the situation

Answer 1

My first thought was he might be arrested because he is in front of Mrs. Brodkey's window!

Using the equation of y = Asin(Bt)

But the period time would be from 3m to 10m and back, which takes 2x3 seconds = 6 seconds.

The period is 2π/B, so B = π/3.

And the radius of the arc has a diameter of 13-3 = 10', so the radius = 5'

Then

y = 5Sin((π/3)t)

Answer 2

It appears that you are being asked to describe the harmonic motion of a pendulum.The general formula for that is (y - k)/b = sin(x - h)/a, where a is the period divided by 2pi, b is 1/2 the distance the pendulum travels back and forth, h is the phase shift (or horizontal shift of the graph), and k is the vertical shift (or the minimum distance plus 1/2 the total distance the pendulum travels).

Given that, the pendulum takes 3 seconds to travel from 3 meters to 13 meters, and it would need 3 seconds to come back to 3 meters again. So the period is 6 seconds, and a = 6/2pi = 3/pi.

The pendulum travels from 3 meters to 13 meters, a distance of 10 meters, so b = 10/2 = 5 meters.

The vertical shift, k, is how far above the x-axis the midpoint of the sine wave function will be. That will be the minimum distance (3 meters) plus one half of the travel distance (5 meters), so k = 8 meters.

The horizontal, or phase shift, describes how far the sine wave function is shifted left or right. For a sine wave, it takes 1/4 the period for the wave to move from a peak or a trough to the midpoint of the wave. The period is 6, so it was at a peak 6 seconds before the data you were give, at t = 2 seconds. With no horizontal shift, the wave would have been at the midpoint of the graph, and crossing the y-intercept, 1/4 the period time prior to the 2 second peak. But since the period is 6 seconds, 1/4 of that is 1.5 seconds, and when the wave crosses the midpoint it is 0.5 seconds past the time it would be crossing the y-intercept, so there is a horizontal shift of 0.5 seconds. Thus, h = 0.5.

Now put this all into the general function, (y - k)/b = sin(x-h)/a to get:

(y-8)/5 = sin((x-0.5)/(3/pi)).

To put it into standard form, first multiply both sides by 5, and then add 8 to both sides to get:

y = 5sin((x-0.5)/(3/pi)) + 8.

Hope this helps.