mike leaves his house for a bike ride. point p is shown on an image of one of mikes bicycle wheels below…
mike leaves his house for a bike ride. point p is shown on an image of one of mikes bicycle wheels below. let $p(t)=8sin(\frac{pi}{2}+\frac{2pi}{3}t)+10$ model the vertical position of point p, in inches, t seconds after mike began his bike ride. part a what is the diameter of mikes bicycle wheels? show or explain your reasoning.
Answer
Explanation:
Step1: Identify the amplitude and vertical - shift
The general form of a sinusoidal function is $y = A\sin(Bx - C)+D$. In the given function $P(t)=8\sin(\frac{\pi}{2}+\frac{2\pi}{3}t)+10$, the amplitude $A = 8$ and the vertical - shift $D = 10$.
Step2: Determine the maximum and minimum values
The maximum value of the sine function $\sin(\theta)$ is 1 and the minimum value is - 1. The maximum value of $P(t)$ is $P_{max}=8\times1 + 10=18$ (when $\sin(\frac{\pi}{2}+\frac{2\pi}{3}t)=1$). The minimum value of $P(t)$ is $P_{min}=8\times(-1)+10 = 2$ (when $\sin(\frac{\pi}{2}+\frac{2\pi}{3}t)=-1$).
Step3: Calculate the diameter
The diameter of the wheel is the difference between the maximum and minimum vertical positions of point $P$. $d=P_{max}-P_{min}=18 - 2=16$ inches.
Answer:
16 inches