Question 1149303
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The height above the ground is a sinusoidal function.<br>
The minimum height is 2; the maximum height is 2+45 = 47; the midline is (2+47)/2 = 24.5.<br>
We can model a revolution of the Ferris wheel as starting at the loading platform, which is the minimum height of the ride; so we can model the height with a negative cosine function:<br>
{{{24.5-22.5*cos(x)}}}<br>
For this problem, we don't need to use the 10-minute rotation time of the Ferris wheel in our function; we can use a "plain" cosine function and simply determine what fraction of one revolution is spent above 28 feet.<br>
Here is a graph of a bit more than one period of the function, along with the constant function 28, showing the ride at its minimum height at 0 degrees of revolution and again at 360 degrees:<br>
{{{graph(400,400,-60,420,-5,50,24.5-22.5*cos((pi/180)x),28)}}}<br>
To find the fraction of a period during which the height of the ride is above 28 feet, you can use a graphing calculator to find the points of intersection of the two graphs.<br>
Algebraically, you can find the two angles when the height of the ride is 28 feet by solving the equation<br>
{{{24.5-22.5*cos(x) = 28}}}
{{{-22.5*cos(x) = 3.5}}}
{{{cos(x) = arccos(-3.5/22.5)}}}<br>
You can find those angles by either of those methods; I leave that to you.<br>
The final answer is that the ride is at or above 28 feet for approximately 162.1 degrees of every 360-degree revolution.<br>
{{{162.1/360 = 0.450}}} to 3 decimal places.<br>
Then, since the period of revolution is 10 minutes, the ride is above 28 feet for 4.5 minutes each revolution.<br>