1. **Problem Statement:** We need to find which equation correctly models the height $H(t)$ of a rider on the London Eye as a function of time $t$ in minutes. 2. **Given Information:** - Radius of the wheel $r = 68$ m. - One complete revolution takes 30 minutes. - Starting height at $t=0$ is 2 m. - The height function is of the form $$H(t) = a \sin|b(t - c)| + d$$ 3. **Understanding the parameters:** - Amplitude $a$ corresponds to the radius of the wheel or half the vertical distance covered. - Vertical shift $d$ corresponds to the center height of the wheel above the ground. - The period $T$ of the sine function is related to $b$ by $$T = \frac{2\pi}{b}$$ - Horizontal shift $c$ adjusts the phase to match the starting height. 4. **Calculate the period and $b$:** - The wheel completes one revolution in 30 minutes, so the period $T = 30$. - Using $$b = \frac{2\pi}{T} = \frac{2\pi}{30} = \frac{\pi}{15}$$ 5. **Determine amplitude $a$ and vertical shift $d$:** - The wheel radius is 68 m, so the amplitude $a$ should be 68. - The lowest point is 2 m, so the center height $d$ is radius + lowest point = $68 + 2 = 70$ m. 6. **Check the horizontal shift $c$:** - The function starts at height 2 m at $t=0$, which is the minimum point. - For a sine function, minimum occurs at phase $\frac{3\pi}{2}$ or by shifting the function by $7.5$ minutes. - So, $c = 7.5$. 7. **Evaluate the options:** - Option A: $a=68$, $b=\frac{\pi}{15}$, $c=-7.5$, $d=68$ (vertical shift too low) - Option B: $a=68$, $b=\frac{\pi}{15}$, $c=7.5$, $d=70$ (matches amplitude, period, shift, and vertical shift) - Option C: $a=34$, $b=\frac{\pi}{30}$, $c=-7.5$, $d=70$ (amplitude and period incorrect) - Option D: $a=34$, $b=\frac{\pi}{30}$, $c=7.5$, $d=68$ (amplitude and vertical shift incorrect) 8. **Conclusion:** The correct equation is Option B: $$H(t) = 68 \sin\left|\frac{\pi}{15}(t - 7.5)\right| + 70$$ --- **Additional question:** Given $f(\theta) = \cos(n\theta)$ has the same period as $g(\theta) = \tan \theta$, find $n$. - Period of $\tan \theta$ is $\pi$. - Period of $\cos(n\theta)$ is $$\frac{2\pi}{n}$$ - Set equal: $$\frac{2\pi}{n} = \pi \Rightarrow n = 2$$ **Final answers:** - Height function: $$H(t) = 68 \sin\left|\frac{\pi}{15}(t - 7.5)\right| + 70$$ - Value of $n$ for cosine function: $2$