Modeling a bouncing thrill ride with a sinusoid

Midline and amplitude. The height runs between 20 and 120 feet: $$\text{midline}=\frac{120+20}{2}=70,\text{amplitude}=\frac{120-20}{2}=50$$

Period. One bounce, from top to bottom and back to the top, takes 8 seconds, so the period is 8. For $\cos (Bt)$ the period is $\frac{2\pi }{B}$, so $$B=\frac{2\pi }{8}=\frac{\pi }{4}$$

Sine or cosine. At $t=0$ the point is at its highest, 120 feet. Cosine starts at its maximum ($\cos 0=1$), while sine starts at its midline. So use cosine with a positive amplitude: $$H(t)=50\cos \left(\frac{\pi }{4}t\right)+70$$ That is choice B.

Check: $H(0)=50+70=120$, $H(4)=50\cos \pi +70=20$ (the bottom, halfway through the bounce), and $H(8)=50\cos 2\pi +70=120$.

Each choice

• A. Sine starts at the midline. This gives $H(0)=70$, so the ride would begin halfway up instead of at the top.
• B (correct). Cosine starts at its maximum, $\frac{2\pi }{8}=\frac{\pi }{4}$ gives a period of 8 seconds, and the midline 70 with amplitude 50 covers 20 to 120 feet.
• C. Sine gives $H(0)=70$ instead of 120, and $B=\frac{\pi }{8}$ makes the period 16 seconds. The 8 seconds was put directly into the denominator rather than into $\frac{2\pi }{B}$.
• D. Cosine is the right choice for starting at the top, but $B=\frac{\pi }{8}$ gives a period of $\frac{2\pi }{\pi /8}=16$ seconds. At $t=8$ this has the point at the bottom, 20 feet, when it should be back at 120.