Hyun Woo is riding a ferris wheel. $H(t)$ models his height (in $\mathrm{m}$ ) above the ground, $t$ seconds after the ride starts. Here, $t$ is entered in radians.
$$
H(t)=-10 \cos \left(\frac{2 \pi}{150} t\right)+10
$$

When does Hyun Woo first reach a height of $16 \mathrm{~m}$ ?
Round your final answer to the nearest whole second.
$\square$ seconds

Question

Hyun Woo is riding a ferris wheel. $H(t)$ models his height (in $\mathrm{m}$ ) above the ground, $t$ seconds after the ride starts. Here, $t$ is entered in radians.
$$
H(t)=-10 \cos \left(\frac{2 \pi}{150} t\right)+10
$$

When does Hyun Woo first reach a height of $16 \mathrm{~m}$ ?
Round your final answer to the nearest whole second.
$\square$ seconds

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Set up the equation to solve for $t$ when $H(t) = 16$: $-10\cos\left(\frac{2\pi}{150}t\right) + 10 = 16$. ‖ ‖ step 2 ⋮ Subtract 10 from both sides of the equation: $-10\cos\left(\frac{2\pi}{150}t\right) = 6$. ‖ ‖ step 3 ⋮ Divide both sides by -10: $\cos\left(\frac{2\pi}{150}t\right) = -\frac{6}{10}$. ‖ ‖ step 4 ⋮ Simplify the right side of the equation: $\cos\left(\frac{2\pi}{150}t\right) = -\frac{3}{5}$. ‖ ‖ step 5 ⋮ Solve for $t$: $t = 150\left(n - \frac{\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)}{\pi}\right)$ and $n \in \mathbb{Z}$. ‖ ‖ step 6 ⋮ Solve for $t$ using the positive value of $n$ that gives the smallest positive $t$: $t = \frac{150\left(\pi n + \sin^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)}{\pi}$ and $n \in \mathbb{Z}$. ‖ ‖ step 7 ⋮ Calculate the value of $t$ for $n=0$: $t = \frac{150\left(\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)}{\pi}$. ‖ ‖ step 8 ⋮ Use a calculator to find the value of $\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$ and multiply by $\frac{150}{\pi}$ to find $t$. ‖ ‖ step 9 ⋮ Round the result to the nearest whole number to find the time in seconds. ‖ ∻[1] Answer∻ ⚹ [Insert final answer here] ⚹ ∻Key Concept∻ ⚹Solving trigonometric equations⚹ ∻Explanation∻ ⚹To find the time when Hyun Woo first reaches a height of 16 meters, we solve the trigonometric equation given by the height function $H(t)$ for $t$. After isolating the cosine function, we use the inverse sine function to find the angle corresponding to the given cosine value, and then solve for $t$ in seconds.◊What is the period of the function $H(t) = -10cos\left(\frac{2\pi}{150}t\right) + 10$ representing Hyun Woo's height above the ground on the Ferris wheel?⍭ Generate me a similar question◊⚹

Sia