A person’s height, in feet above the ground, on a ferris wheel can be modeled using the equation:
h(t)=-45cos(pi t/7)+52
a) Determine the period and describe what the period represents in this context.
b) How long does it take for the rider to reach maximum height?
c) What changes when the Ferris wheel has a larger radius?
a) The period of the function h(t) is 14. The period represents the time it takes for the ferris wheel to make one full rotation around its center. In this context, it represents the time it takes for the rider to go from their starting height, to the highest point on the ferris wheel, and back to their starting height.
b) To find when the rider reaches maximum height, we need to find the time when the cosine function reaches its maximum value. This occurs when the argument of the cosine function, pi t/7, is equal to 0. This gives us t=0 and t=14 as the times when the rider reaches maximum height. Therefore, it takes 14 units of time for the rider to reach maximum height.
c) When the ferris wheel has a larger radius, the amplitude of the cosine function (45 in this case) would increase. This means that the variation in the rider's height above the ground would be larger, and they would experience greater changes in height as the ferris wheel rotates. Additionally, the maximum height that the rider reaches would be higher with a larger radius ferris wheel.
can you explain part b more
Of course! In the given equation h(t) = -45cos(pi t/7) + 52, the function represents the height of the rider above the ground at any given time t. The cosine function oscillates between -1 and 1, which means the maximum height that the rider can reach is given by -45 * -1 + 52 = 97 and the minimum height is given by -45 * 1 + 52 = 7.
To find when the rider reaches maximum height, we need to find the values of t that correspond to the peaks of the cosine function. Since the cosine function has a period of 14 units (2π divided by the coefficient of t, 7 in this case), we see that the maximum height occurs at t = 0 and t = 14.
At t = 0, the height is given by h(0) = -45cos(0) + 52 = -45 * 1 + 52 = 7, which is the minimum height.
At t = 14, the height is given by h(14) = -45cos(2π) + 52 = -45 * 1 + 52 = 7, which is the minimum height.
So, it takes 14 units of time for the rider to go from the minimum height, reach the maximum height, and then return to the minimum height again.