QuestionA ferris wheel is 16 meters in diameter and makes one revolution every 8 minutes. For how many minutes of any revolution will your seat be above 12 meters?
For minutes of any one revolution you will be above 12 meters.
Studdy Solution
STEP 1
What is this asking?
How long are we higher than 12 meters on a 16-meter diameter Ferris wheel that takes 8 minutes for one full spin?
Watch out!
Don't mix up radius and diameter!
Also, remember the Ferris wheel is a circle, so trigonometry might be helpful!
STEP 2
1. Set up the Ferris wheel's height function.
2. Find the angles where the height is 12 meters.
3. Calculate the time spent above 12 meters.
STEP 3
Let's **define the function** \(h(t)\) that gives our height at time .
We know the **diameter** is 16 meters, so the **radius** is \(16/2 = 8\) meters.
STEP 4
Imagine the Ferris wheel centered at in the coordinate plane.
Our height at any time can be modeled using cosine, since we start at the top.
The **height function** is \(h(t) = 8\cos(\omega t) + 8\), where 8 is the **radius**, is the **angular speed**, and the shifts the center up to 8 meters.
STEP 5
The wheel takes 8 minutes for one full rotation, which is radians.
So, the **angular speed** is \(2\pi / 8 = \pi/4\) radians per minute.
Our **height function** becomes \(h(t) = 8\cos\left(\frac{\pi}{4}t\right) + 8\).
STEP 6
We want to find the times when \(h(t) = 12\).
So, let's **solve** \(12 = 8\cos\left(\frac{\pi}{4}t\right) + 8\).
STEP 7
**Subtract** 8 from both sides: \(12 - 8 = 8\cos\left(\frac{\pi}{4}t\right) + 8 - 8\), which simplifies to \(4 = 8\cos\left(\frac{\pi}{4}t\right)\).
STEP 8
**Divide** both sides by 8: \(4/8 = 8\cos\left(\frac{\pi}{4}t\right)/8\), giving us \(1/2 = \cos\left(\frac{\pi}{4}t\right)\).
STEP 9
The angles where cosine is \(1/2\) are \(\pi/3\) and \(5\pi/3\) in one rotation.
So, \(\frac{\pi}{4}t = \frac{\pi}{3}\) and \(\frac{\pi}{4}t = \frac{5\pi}{3}\).
STEP 10
**Multiply** both sides of the first equation by \(4/\pi\): \(\frac{4}{\pi} \cdot \frac{\pi}{4}t = \frac{4}{\pi} \cdot \frac{\pi}{3}\), which gives us \(t = 4/3\) minutes.
STEP 11
**Multiply** both sides of the second equation by \(4/\pi\): \(\frac{4}{\pi} \cdot \frac{\pi}{4}t = \frac{4}{\pi} \cdot \frac{5\pi}{3}\), which gives us \(t = 20/3\) minutes.
STEP 12
The time between and is the time spent above 12 meters.
STEP 13
**Subtract** the two times: \(20/3 - 4/3 = 16/3\) minutes.
STEP 14
You will be above 12 meters for \(16/3\) minutes of any one revolution.
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