To sketch the graph representing the height of a rider above the ground as a function of time, we can start by determining the equation for the height.
The diameter of the Ferris wheel is given as 56 m, which means the radius is 28 m (diameter/2). The circumference of the wheel can be found using the formula:
Circumference = π * Diameter = π * 56m = 176 m
Since one revolution takes 2.5 minutes, the period of the function is 2.5 minutes or 150 seconds. The equation for the height of the rider above the ground, as a function of time, can be written as:
h(t) = A * sin((2π * t) / T)
Where:
h(t) = height above the ground at time t
A = amplitude, which is equal to the radius (28 m in this case)
t = time
T = period, which is 150 seconds in this case
Given that the rider gets on at a height of 0.5 m at t=0 min, we can modify the equation to incorporate this initial condition:
h(t) = A * sin((2π * t) / T) + 0.5
Now let's determine the intervals when the rider could see Niagara Falls. The rider can see Niagara Falls if the height above the ground (h(t)) is greater than 50 m.
So, we need to solve the inequality:
A * sin((2π * t) / T) + 0.5 > 50
Now, let's sketch the graph for the height of the rider above the ground as a function of time and identify the intervals when the rider could see Niagara Falls.
(Note: Since it's difficult to accurately sketch a graph using plain text, I can describe how the graph would look like)
Graph Sketch:
1. The graph is a sinusoidal wave that starts at 0.5 m (the initial height).
2. The maximum height is A + 0.5 m, which is 28 + 0.5 = 28.5 m.
3. The minimum height is -A + 0.5 m, which is -28 + 0.5 = -27.5 m.
4. The period is 150 seconds, and the graph completes three cycles within this period.
5. The values of t for the three complete cycles will be between 0 and 450 seconds.
Intervals when the rider could see Niagara Falls:
Based on the graph, the rider could see Niagara Falls when the height is above 50 m. From the graph sketch, this occurs during the crest portions of the cycle.
Therefore, the time intervals when the rider could see Niagara Falls are the intervals where the graph is above 50 m.