(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.