Question

Math

Posted 3 months ago

```
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
```

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Answer from Sia

Posted 3 months ago

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.

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