Question

fill in the blank 25 points
a ferris wheels rotation can be modeled by this function where t is time in seconds since the start of the ride and h(t) represents the height of the ferris wheel. at what time will a rider be 25 feet above the ground?
h(t)=50 sin(\frac{2\pi}{60}t)
t =

Explanation:

Step1: Substitute \(h(t) = 25\) into the function

We have the equation \(25=50\sin(\frac{2\pi}{60}t)\).

Step2: Simplify the equation

Divide both sides by \(50\): \(\frac{25}{50}=\sin(\frac{\pi}{30}t)\), so \(\sin(\frac{\pi}{30}t)=\frac{1}{2}\).

Step3: Solve for \(t\) using the inverse - sine function

We know that if \(\sin(x)=\frac{1}{2}\), then \(x = 2k\pi+\frac{\pi}{6}\) or \(x=(2k + 1)\pi-\frac{\pi}{6}\), \(k\in\mathbb{Z}\).

For \(x=\frac{\pi}{30}t\):

• Case 1: \(\frac{\pi}{30}t=2k\pi+\frac{\pi}{6}\)

Multiply both sides by \(\frac{30}{\pi}\): \(t = 60k + 5\).

• Case 2: \(\frac{\pi}{30}t=(2k + 1)\pi-\frac{\pi}{6}\)

\(\frac{\pi}{30}t=2k\pi+\pi-\frac{\pi}{6}=2k\pi+\frac{5\pi}{6}\)
Multiply both sides by \(\frac{30}{\pi}\): \(t=60k + 25\).

Since \(t\geq0\), when \(k = 0\):

• From \(t = 60k+5\), we get \(t = 5\) seconds.
• From \(t=60k + 25\), we get \(t = 25\) seconds.

Answer:

\(t = 5\) seconds or \(t = 25\) seconds