trigonometry word problem

Question

Terence is enjoying an ice cream cone as he rides on a Ferris wheel that has a diameter of 65 feet. The Ferris wheel is turning at a constant rate. Terence notices that it takes 7 minutes to go from the bottom of the ride to the top of the ride. At the bottom of the ride he is 13 feet above the ground. At the top of the ride Terence suddenly drops his ice cream cone. Find a formula for Terence’s height above the ground as a function of the time (in minutes) since he dropped his ice cream.

Answer 1

The formula (generally) for a trigonometric function can be modeled by: y = (+/-) A*f(Bt) + C, where A represents the amplitude of the function, B is related to the period of the function, and C is the midline equation of the function.

C - The midline of the function is found halfway between the highest and lowest points of the function. The lowest point of the Ferris wheel is 13 feet above the ground, while the highest point is 13 + 65 = 78 feet off the ground. Therefore, the midline of the function is found by C = 0.5*(HI+LO) = 0.5*(78+13) = 45.5 feet

A - The amplitude of the function is found by the difference between the highest point and the midline of the function. Here, A = HI - ML = 78 - 45.5 = 32.5

B - The period of the function represents the time it takes for one complete cycle of the function to occur. The problem states it takes 7 minutes for a one-half complete revolution of the Ferris wheel, therefore the period is equal to 14 minutes. The value of B in the function is found using the formula: B = (2π)/(Period), or B = (2π)/14, or B = π/7.

f(t) - The problem states that at the top of the Ferris wheel, Terrance loses his ice cream cone, and we are asked to identify the type of function from this point. A cosine function begins its maximum value, and since Terrance loses his cone at the top of the Ferris wheel, we know that this function can be modeled by a cosine graph.

Therefore, the function that models Terrance's height, f(t), at any time t after dropping his ice cream cone, can be modeled by: f(t) = 32.5*cos((π/7)*t)+45.5