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Question
Math
Posted 3 months ago

Hyun Woo is riding a ferris wheel. H(t)H(t) models his height (in m\mathrm{m} ) above the ground, tt seconds after the ride starts. Here, tt is entered in radians.
H(t)=10cos(2π150t)+10
H(t)=-10 \cos \left(\frac{2 \pi}{150} t\right)+10


When does Hyun Woo first reach a height of 16 m16 \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 tt when H(t)=16H(t) = 16: 10cos(2π150t)+10=16-10\cos\left(\frac{2\pi}{150}t\right) + 10 = 16
step 2
Subtract 10 from both sides of the equation: 10cos(2π150t)=6-10\cos\left(\frac{2\pi}{150}t\right) = 6
step 3
Divide both sides by -10: cos(2π150t)=610\cos\left(\frac{2\pi}{150}t\right) = -\frac{6}{10}
step 4
Simplify the right side of the equation: cos(2π150t)=35\cos\left(\frac{2\pi}{150}t\right) = -\frac{3}{5}
step 5
Solve for tt: t=150(nsin1(25)π)t = 150\left(n - \frac{\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)}{\pi}\right) and nZn \in \mathbb{Z}
step 6
Solve for tt using the positive value of nn that gives the smallest positive tt: t=150(πn+sin1(25))πt = \frac{150\left(\pi n + \sin^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)}{\pi} and nZn \in \mathbb{Z}
step 7
Calculate the value of tt for n=0n=0: t=150(sin1(25))πt = \frac{150\left(\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)}{\pi}
step 8
Use a calculator to find the value of sin1(25)\sin^{-1}\left(\frac{2}{\sqrt{5}}\right) and multiply by 150π\frac{150}{\pi} to find tt
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)H(t) for tt. 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 tt in seconds.

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