Question

anastelles bicycle has a wheel radius of 33 inches. she places a sticker on the wheel so that its minimum height above the ground is 0.5 inches. when she rides her bicycle, the wheel completes 30 revolutions every minute. the sticker begins at its minimum height above the ground. which equation models the height in inches of the sticker after x minutes? y = 0.5 sin(180πx)+33 y = 12.5 sin(180πx)+33 y = 0.5 sin(180πx - π/2)+33 y = 12.5 sin(180πx - π/2)+33

Explanation:

Step1: Determine the amplitude

The radius of the wheel is $r = 13$ inches and the minimum height of the sticker above the ground is $0.5$ inches. The center - height of the wheel above the ground is $h=13 + 0.5=13.5$ inches. The amplitude $A$ of the sine - wave is the distance from the center - height to the maximum or minimum height. The maximum height of the sticker above the ground is $13\times2 + 0.5=26.5$ inches. The amplitude $A=\frac{26.5 - 0.5}{2}=13$. But we can also calculate it as $A = 13 - 0.5=12.5$.

Step2: Determine the vertical shift

The vertical shift $D$ is the height of the center of the wheel above the ground, so $D = 13+0.5 = 13$.

Step3: Determine the angular frequency

The wheel completes $30$ revolutions per minute. One revolution is $2\pi$ radians. So the angular frequency $\omega$ (in radians per minute) is $\omega=30\times2\pi = 60\pi$ radians per minute. But if we consider the general form of a sinusoidal function $y = A\sin(Bx - C)+D$, and we know that for a periodic function $y = A\sin(Bx - C)+D$ with period $T$, $\omega = B$. Since the wheel rotates $30$ times in $1$ minute, and the general form of the sine function for this problem has $x$ in minutes, $B = 60\pi$.

Step4: Determine the phase - shift

The sticker starts at its minimum height. For a sine function $y = A\sin(Bx - C)+D$, when $x = 0$, $y$ is at its minimum. The standard sine function $y=\sin x$ has a minimum at $x=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}$. For $y = A\sin(Bx - C)+D$, when $x = 0$, $Bx - C=-\frac{\pi}{2}$. Since $B = 60\pi$ and $x = 0$, $C=\frac{\pi}{2}$.
The general form of the sinusoidal function for the height $y$ of the sticker after $x$ minutes is $y=A\sin(Bx - C)+D$. Substituting $A = 12.5$, $B = 60\pi$, $C=\frac{\pi}{2}$, and $D = 13$ we get $y = 12.5\sin(60\pi x-\frac{\pi}{2})+13$. But if we rewrite the function using the identity $\sin(a - b)=\sin a\cos b-\cos a\sin b$, we can also note that the function can be written in terms of the number of revolutions. Since the wheel rotates $30$ times per minute, and we know that the general form of a sine - wave for this problem is $y = A\sin(Bx - C)+D$. The correct function is $y = 12.5\sin(60\pi x-\frac{\pi}{2})+13$.

Answer:

$y = 12.5\sin(60\pi x-\frac{\pi}{2})+13$ (corresponding to the option that has $y = 12.5\sin(180\pi x-\frac{\pi}{2})+13$ assuming there is a typo in the problem - statement where the angular frequency might be mis - written as $180\pi$ instead of $60\pi$ in the options, and among the given options the closest correct one is the one with $y = 12.5\sin(180\pi x-\frac{\pi}{2})+13$)