Question

Jami is trying to model the movement of a tire. In order to properly model the movement, she is going to focus on the movement of the tire's air nozzle. The nozzle is on the right side of the tire and she is going to treat that position as the equilibrium position. She then slowly rotates the tire so that the nozzle goes up at the beginning of the rotation and follows its change in height. The distance of the nozzle from the center of rotation is nine inches. She finds that it takes eight seconds for the tire to make a single revolution. Which of the following functions best models the position of the nozzle?

A) s(t) = 9cos(t)
B) s(t) = 9sin(t)
C) s(t) = 9sin(47)
D) s(t) = 9cos(47t)

Answer 1

The best function to model the nozzle's position on a tire that rotates every eight seconds with a nine-inch radius is s(t) = 9 cos(π/4 t), which is not exactly matched by any of the given options.

Step-by-step explanation:

Jami is trying to model the movement of a tire by focusing on the movement of the tire's air nozzle, which she considers to be at the equilibrium position when on the right side of the tire. Given that the distance from the center of rotation to the nozzle is nine inches and the tire makes a single revolution every eight seconds, we need a trigonometric function that represents this periodic motion and takes into account the period of rotation.

The general form of the function to model periodic motion is s(t) = A cos(2πft + ϕ) or s(t) = A sin(2πft + ϕ), where A is the amplitude, f is the frequency, t is the time, and ϕ is the phase shift. Here, the amplitude is 9 inches, and the frequency is the reciprocal of the period (1/8 revolutions per second), which needs to be converted to radians per second for the function. The function will also need to start at the maximum, since that's the initial position of the nozzle.

Thus, the function that best models the position of the nozzle is:

s(t) = 9 cos(π/4 ⋅ t), which aligns with none of the given options exactly. Therefore, we would advise Jami to consider this newly derived function that correctly accounts for the amplitude, period, and initial condition of the nozzle's motion.