A ferris wheel has a radius of 13 m. It rotates once every 24 seconds. A passenger gets on at the bottom
of the wheel from a ramp which is one metre above ground level.
a) If the height of the passenger is measured from the ground, determine an equation for the height of the
passenger as a function of time in the form of h(t)= acos(bt)+d
b) To the nearest metre, find the height of the passenger after 55 seconds.
2π/k = period
2π/k = 24
k = 2π/24 = π/12 <---- that's the b of your equation.
also we know a = 13
so we start with h(t) = 13 cos (π/12 t)
the minimum of this graph is -13, we want the min to be +1, so we have to raise it 14 units
h(t) = 13 cos (π/12 t) + 14
I would have started with a sine function, rather than a cosine function, since the sine starts at 0 when t = 0 and would be increasing.
The cosine curve would start at 1 when t = 0
So we have to move our cosine function horizontally to achieve this.
Let's see what we have so far
http://www.wolframalpha.com/input/?i=h(t)+%3D+13+cos+(%CF%80%2F12+t)+%2B+14
if we translate our curve 12 units to the right, we get:
http://www.wolframalpha.com/input/?i=h(t)+%3D+13+cos+((%CF%80%2F12)(t+-+12)))+%2B+14
so h(t) = 13cos ( (π/12)(t-12) ) + 14
checking:
when t = 0 , h(t) = 13 cos (-π) + 14 = 1 --> bottom
when t = 6 , h(6) = 13cos(-π/2)+14 = 14 --> half-way up
when t = 12, h(12)=13cos(0)+14 = 27 --> the top
when t= 18, h(18) =13cos(π/2)+14 = 14
when t=24, h(24) = 1
equation is good,
b) when t = 55
wheel has gone 55/24 = 2 + 7/24 periods
so it would be in the same position as it would be at 7 seconds
h(7) = 13cos (-5π/12) + 14
= appr 17.4 m high
To determine the equation for the height of the passenger as a function of time, we need to consider the motion of the ferris wheel. Let's break down the problem into smaller steps:
Step 1: Find the equation for the position of a point on the ferris wheel as a function of time.
The position of a point on the ferris wheel can be described using trigonometric functions. Since the ferris wheel has a radius of 13 m and completes one rotation every 24 seconds, we can use the equation:
x(t) = r * cos(2π * t / T)
where x(t) is the horizontal position, r is the radius of the ferris wheel (13 m), t is the time, and T is the period of rotation (24 seconds).
Step 2: Adjust the equation to calculate the height of the passenger.
Since the passenger gets on the ferris wheel from a ramp one meter above ground level, we need to account for this offset in the equation. The height (h) can be obtained by adding the radius of the ferris wheel (13 m) and the additional one-meter offset:
h(t) = r * cos(2π * t / T) + 1
Step 3: Simplify the equation.
To match the given form h(t) = acos(bt) + d, we can simplify the equation by choosing appropriate values for a, b, and d. Let's set a = r, b = (2π / T), and d = 1. The equation becomes:
h(t) = 13 * cos((2π * t) / 24) + 1
Therefore, the equation for the height of the passenger as a function of time is h(t) = 13 * cos((π / 12) * t) + 1.
To find the height of the passenger after 55 seconds, we can substitute t = 55 into the equation:
h(55) = 13 * cos((π / 12) * 55) + 1
Using a calculator, the value of h(55) is approximately 11 meters. Hence, to the nearest meter, the height of the passenger after 55 seconds is 11 meters.