A Ferris wheel with a diameter of 100
feet rotates at a constant rate of 4 revolutions per minute. Let the center of the Ferris
wheel be at the origin.
The height h (in feet) of a Ferris wheel car located at the point (x, y) is given by
h = 50 + y
where y is related to the angle è (in radians) by the equation
y = 50 sin è
(a) Write an equation of the height h in terms of time t (in minutes). (Hint: One revolution is 2ð radians. y = 50 sin è
revolution is 2 radians.)
(b) Sketch a graph of the equation you wrote in part (a).
(c) Use the verical line test to determine whether h is a function of t.
(d) What does your answer to part (c) mean in the context of the problem?
h = 50 + 50sinθ
we all know sinθ is a function
so, at any angle θ there is only one height the car can have.
Which on can you please help me
(a) To write the equation of height h in terms of time t, we can relate the angle è to time t using the given constant rate of rotation.
Since the Ferris wheel rotates at a constant rate of 4 revolutions per minute, the angle è in radians can be calculated as:
è = (2π radians/revolution) * (4 revolutions/minute) * t
Substituting this value of è into the equation y = 50 sin è, we get:
y = 50 sin[(2π radians/revolution) * (4 revolutions/minute) * t]
Finally, substituting y into the equation h = 50 + y, we get the equation for the height h in terms of time t:
h = 50 + 50 sin[(2π radians/revolution) * (4 revolutions/minute) * t]
(b) To sketch the graph of the equation, we can plot the height h on the y-axis and time t on the x-axis.
(c) To determine whether h is a function of t, we can use the vertical line test. This test states that if a vertical line intersects the graph at more than one point, then the graph does not represent a function.
(d) If the vertical line test shows that h is not a function of t, it means that for a certain time t, there are multiple different heights for the Ferris wheel car located at the point (x, y). This implies that the car may have different positions at the same time, which is not possible.
To answer the given questions, we need to go step by step. Let's start with part (a), where we have to write an equation of the height `h` in terms of time `t`.
(a) We are given that the Ferris wheel rotates at a constant rate of 4 revolutions per minute. One revolution is equal to 2π radians. Let's convert it to minutes:
1 revolution = 2π radians
4 revolutions = 4 * 2π radians
Now, let's relate the angle `θ` in radians to time `t` in minutes. Since the Ferris wheel rotates at a constant rate, we can say:
θ = (4 * 2π * t) / 1
Using the given relation, `y = 50 sin θ`, we can substitute the value of `θ` to get:
y = 50 sin ((4 * 2π * t) / 1)
Finally, we can relate `y` to `h` as `h = 50 + y`. Substituting the value of `y`, we get the equation of height `h` in terms of time `t`:
h = 50 + 50 sin ((4 * 2π * t) / 1)
(b) To sketch a graph of the equation h = 50 + 50 sin ((4 * 2π * t) / 1), we need to plot the height `h` on the y-axis and time `t` on the x-axis. The graph will be sinusoidal with a maximum height of 100 feet (50 + 50) and a minimum height of 0 feet (50 - 50). The period of the function will be 1/4 minutes.
(c) To determine whether `h` is a function of `t`, we can use the vertical line test. The vertical line test states that if every vertical line intersects the graph at most once, then the graph represents a function. In this case, the vertical line test is satisfied as each vertical line intersects the graph at most once. Therefore, `h` is a function of `t`.
(d) In the context of the problem, the answer to part (c) means that for each given time `t`, there is a unique corresponding height `h`. As the Ferris wheel rotates, the height of the car changes, and for each specific time, there is a single height associated with it. This reinforces the fact that the height of the car is dependent on the time and can be represented as a function of `t`.