Question 1189142: the height, h in meters, above the ground of a rider on a ferris wheel can be modelled by the equation h=8sin(π/15(t-5.5))+13, where t is the time in seconds. at t=0 seconds, the rider is at the lowest point. determine the first two times that the rider is 17m above the ground.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the general equation of the sine wave is:
y = a * sin(b * (x - c)) + d
a is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift.
the equation is:
h = 8 * sin(pi/15 * (t - 5.5)) + 13.
the amplitude is 8 and the vertical shift is 13.
that means the horizontal center line of the equation is at h = 13 and the high point of the sine wave is at 13 + 8 = 21 and the low point of the sine wave is at 13 - 8 = 5.
the normal period of a sine wave is 2pi.
since the frequency is pi/15, then the period will be 2pi / (pi/15) = 2pi * 15 / pi = 30.
that means one cycle of the sine wave will have a width of 30 units.
since t is in seconds, then one full cycle of the sine wave will take place in 30 seconds.
the horizontal shift is represented by (t - 5.5) which means the sine wave is shifted to the right by 5.5 seconds.
that means the start of the sine wave will be at 5.5 seconds rather than at 0 seconds.
the problem states that the sine wave will be at its lowest point when t = 0.
when t = 0, the equation becomes h = 8 * sin(pi/15 * (0 - 5.5)) + 13.
simplify that to get:
h = 8 * sin(pi/15 * -5.5) + 13.
simplify further to get:
h = 5.691636339.
that is NOT the lowest point.
the lowest point is at t = -2
that means the horizontal shift has to be 7.5 units to the right and NOT 5.5 units to the right.
when the horizontal shift is 7.5 units to the right, then you get h = 5 when t = 0, which is correct as the lowest point of travel of the sine wave.
bottom line is you have a bad equation that is not tracking with what is stated as the lowest point when t = 0.
i will assume the equation needs to be h = 8 * sin(pi/15 * (t - 7.5) + 13, and answer the questions accordingly.
you can solve for when the low point needs to be by making h = 5 and solving the equation.
you will get:
5 = 8 * sin(pi/15 * (t - 7.5)) + 13.
subtract 13 from both sides to get:
-8 = 8 * sin(pi/15 * (t - 7.5))
divide both sides by 8 to get:
-1 = sin(pi/15 * (t - 7.5))
solve for pi/15 * (t - 7.5) by taking the arcsin of -1 to get:
arcsin(-1) = pi/15 * (t - 7.5)
since arcsin(-1) = -1.570796327, you get:
-1.570796327 = pi/15 * (t - 7.5)
multiply both sides of this equation by 15/pi to get:
15/pi * -1.570796327 = t - 7.5
simplify to get:
-7.5 = t - 7.5
solve for t to get:
t = 0
that is correct and is now in line with what is stated in the problem statement.
you are asked to find the value of t when h = 17.
you solve for that in the same way by making h = 17 and solving for t.
the equation becomes:
17 = 8 * sin(pi/15 * (t - 7.5) + 13
subtract 13 from both sides of the equation to get:
4 = 8 * sin(pi/15 * (t - 7.5)
divide both sides by 8 to get:
.5 = sin(pi/15 * (t - 7.5)
solve for pi/15 * (t - 7.5) by tking the arcsin of .5 to get:
arcsin(.5) = pi/15 * (t - 7.5)
simplify to get:
.5235987756 = pi/15 * (t - 7.5)
multiply both sides by 15/pi to get:
15/pi * .5235987756 = t - 7.5.
simplify to get:
2.5 = t - 7.5
solve for t to get:
t = 10.
that's the first time h = 17.
h = 17 is between the low point of 5 and the high point of 21.
the full cycle is 30.
that means the low point will be at t = 0 and t = 30
the half cycle is when h = 21 since this sine wave starts at h = 0, risees to h = 21, and then drops back down to h = 0.
that means the high point is at t = 15.
since the sine wave is symmetric, and since h = 17 is at t = 10, then h will be 17 again when t = 15 + 5 = 20.
this can also be confirmed by using the formula again to get:
h = 8 * sin(pi/15 * (t - 7.5) + 13 becomes:
h = 8 * sin(pi/15 * (20 - 7.5) + 13 which becomes:
h = 8 * sin(pi/15 * 12.5) + 13 which becomes:
h = 8 * .5 + 13 which becomes:
h = 17.
that confirms that h = 17 when t = 10 and 20.
i also graphed the equation by letting y = h and x = t to get:
y = 8 * (sin(pi/15 * (x - 7.5)) + 13
the graph is shown below:
bottom line is that your equation was not correct.
tince the problem is not that easy to solve when you have the right equation, it did not help that you had the wrong equation.
hopefully the right equation will also lead you to the correct answer as i gave you above.
let me know if you have any questions
theo.
|
|
|
