Explanation:
To find C and D, we can use the given information about the initial position of the person on the ride.
When the person starts the ride, they are 10 feet off the ground. This means that the cosine function has a vertical shift of 10 units, so we have:
f(x) = Acos(Bx - C) + D = 60cos(π/2x - C) + D
At the start of the ride, when x = 0, f(x) = 10. Substituting these values, we get:
10 = 60cos(-C) + D
Simplifying, we get:
D = 10 - 60cos(-C)
We can also use the fact that the minimum height of the ride is 4 feet above the ground. This means that the cosine function has a vertical shift of 4 units, so we have:
f(x) = Acos(Bx - C) + D = 60cos(π/2x - C) + D
At the lowest point of the ride, when x = 1/4, f(x) = 4. Substituting these values, we get:
4 = 60cos(π/8 - C) + D
Substituting D = 10 - 60cos(-C) from the first equation, we get:
4 = 60cos(π/8 - C) + 10 - 60cos(-C)
Simplifying, we get:
cos(-C) = (4 - 10 - 60cos(π/8 - C))/(-60)
cos(-C) = (3cos(π/8 - C) - 1)/2
Using the identity cos(-x) = cos(x), we can rewrite this as:
cos(C) = (3cos(π/8 - C) - 1)/2
Solving for C numerically, we get:
C ≈ 0.438
Substituting this value of C and D = 10 - 60cos(-C) into the equation for f(x), we get:
f(x) = 60cos(π/2x - 0.438) + 10 + 60cos(0.438)