Question

Dominic takes a trip to an amusement park and rides a Ferris wheel. The height, in feet above the ground, of his car over time tt, measured in minutes, is given by h, of, t, equals, 166, point, 4h(t)=166.4 cosleft parenthesis, start fraction, pi, divided by, 3, end fraction, left parenthesis, t, plus, 0, point, 5, right parenthesis, right parenthesis, plus, 185, point, 9(
3
π
​
(t+0.5))+185.9.
What is the minimum and what does it represent in this context?

Answer 1

The cosine ranges from -1 to 1, so the minimum height is
h_min = 185.9 − 166.4 = 19.5 feet.

Interpretation: 19.5 ft is the lowest the Ferris‑wheel car gets above the ground (the bottom of the wheel). This minimum occurs when cos((π/3)(t+0.5)) = −1, i.e. (π/3)(t+0.5) = π + 2πk, so at times t = 2.5 + 6k minutes (first occurrence at t = 2.5 min).