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?

(a) To write an equation of the height h in terms of time t, we need to find the relationship between angle è and time t.

One revolution is equivalent to 2π radians according to the hint. Since the Ferris wheel completes 4 revolutions per minute, the angle è can be related to time t in minutes by the equation:

è = 2π * t * 4

Substituting this value of è into the equation for y, we get:

y = 50 sin(2π * t * 4)

Finally, substituting this value of y into the equation for h, we obtain:

h = 50 + y = 50 + 50 sin(2π * t * 4)

So, the equation for the height h in terms of time t is:
h = 50 + 50 sin(8πt)

(b) Sketching a graph of the equation h = 50 + 50 sin(8πt) would require a visual or plotting tool. Please refer to the graphing software or a graphing calculator to obtain the graph.

(c) To determine whether h is a function of t, we can use the vertical line test. This test states that a relation is a function if any vertical line intersects the graph at most once.

(d) If the vertical line test confirms that h is a function of t, it means that for each value of t, there is only one corresponding value of h. In the context of the problem, it implies that at any given time, there is only one height of the Ferris wheel car.

(a) To write an equation of the height h in terms of time t, we need to find the relationship between the angle è and time t.

Given that the Ferris wheel rotates at a constant rate of 4 revolutions per minute, we can calculate the angular velocity (ω) using the formula:

angular velocity (ω) = (number of revolutions) * (2π radians / 1 revolution) / (time in minutes)

In this case, the number of revolutions is 4, so:

ω = 4 * (2π radians / 1 revolution) / (1 minute) = 8π radians / minute

Now, let's find the relationship between the angle è and time t. We know that one revolution is equal to 2π radians. Therefore, in terms of time t, we have:

è = (angular velocity) * (time) + initial angle

Since the initial angle is not given, we can assume it to be zero (the starting position).

è = 8πt

Now, we can substitute this value of è into the equation for y:

y = 50 sin(è) = 50 sin(8πt)

Finally, substitute y = 50 sin(8πt) into the equation for h:

h = 50 + y = 50 + 50 sin(8πt) = 50(1 + sin(8πt))

Therefore, the equation for the height h in terms of time t is h = 50(1 + sin(8πt)).

(b) To sketch a graph of the equation h = 50(1 + sin(8πt)), we plot the height h on the vertical axis and the time t on the horizontal axis. The graph will be a sinusoidal wave centered at h = 50, with an amplitude of 50.

(c) Using the vertical line test, we can determine whether h is a function of t. The vertical line test states that if any vertical line intersects a graph at more than one point, then the graph does not represent a function.

In the case of the graph of h = 50(1 + sin(8πt)), every vertical line will intersect the graph at most once. Therefore, h is a function of t.

(d) The fact that h is a function of t means that for every value of t (time), there is a unique value of h (height). In the context of the problem, it implies that at any given moment in time, there is only one height at which a Ferris wheel car is located.