the maximun height of a ferris wheel is 35m. the wheel takes 2 mins to make one revolution. passengers board the ferris wheel 2m above the ground at the bottom of the rotation.
a. write an equation to model this
b. how high is the passengers after 45s
c. draw a graph
a. To model the height of the passengers on the Ferris wheel, we can use a cosine function since the Ferris wheel goes up and down in a periodic motion. The general equation for the height of the Ferris wheel at time t is:
h(t) = A * cos(B * t) + C
Where:
A = amplitude of the cosine function, which is half of the total difference between the highest and lowest points, so A = 35 m / 2 = 17.5 m
B = the period of the cosine function, which is 2π divided by the time it takes for one full revolution, so B = 2π / (2 * 60) = π / 60
C = vertical shift of the cosine function, which is the initial height of the passengers, so C = 2 m
Therefore, the equation to model this is:
h(t) = 17.5 * cos((π/60) * t) + 2
b. To find the height of the passengers after 45 seconds, substitute t = 45 into the equation and calculate:
h(45) = 17.5 * cos((π/60) * 45) + 2
Calculating this expression gives:
h(45) ≈ 17.5 * cos(3π/4) + 2
h(45) ≈ -10.57 + 2
h(45) ≈ -8.57 m
Therefore, after 45 seconds, the passengers are approximately 8.57 meters below the initial height (2 meters above the ground).
c. To draw a graph, we can plot the height (h) on the vertical axis and the time (t) on the horizontal axis. The range of t can be from 0 to 120 seconds (assuming one full revolution takes 2 minutes = 120 seconds).
The graph will show the height of the passengers at different times, with the height varying between 17.5 m above the initial height (at the top of the Ferris wheel) and 17.5 m below the initial height (at the bottom of the Ferris wheel).
Please note that as an AI text-based bot, I am unable to draw a visual graph. However, you can use a graphing calculator or a software (such as Microsoft Excel or Desmos) to plot the graph using the equation: h(t) = 17.5 * cos((π/60) * t) + 2
a. To model the height of the passengers on the Ferris wheel, we can use a trigonometric function, specifically a cosine function. Let's assume the angle of rotation is measured in radians.
The equation to model the height of the passengers can be written as:
h(t) = A cos(B(t - C)) + D
Where:
- h(t) represents the height of the passengers above the ground at time t.
- A represents the amplitude, which is half the difference between the maximum and minimum values of the cosine function. In this case, A = 17.5m (half of the maximum height of 35m).
- B represents the frequency, which is the number of complete cycles the cosine function completes in one unit of time. Since the wheel takes 2 minutes (120 seconds) to make one full revolution, we have B = (2π / 120) radians/second.
- C represents the phase shift, which is the horizontal displacement of the cosine function from the usual position. In this case, since passengers board the Ferris wheel 2m above the ground at the bottom of the rotation, we have C = π/2 radians (90 degrees).
- D represents the vertical shift, which is the average value added to the cosine function. In this case, D = 2m (the starting height above the ground).
b. To find the height of the passengers after 45 seconds, we substitute t = 45 seconds into the equation h(t) and solve:
h(45) = 17.5 cos((2π / 120) × (45 - π/2)) + 2
Simplifying this equation will give you the height of the passengers after 45 seconds.
c. To draw a graph, you can use graphing software or online graphing tools. Plot the height (h) on the y-axis and time (t) on the x-axis. Use the equation h(t) from part a to plot points on the graph. Join these points to obtain the graph of the Ferris wheel's height.
The diameter = 35-2 = 33
In 45 seconds, the passenger has traveled 45/120 of its circumference.
I hope this gives you a start.