Ferris Wheel Problem

Today we learned how to create and solve an inverse cosine equation from a word problem. This is the example we did in class.

Here is the problem:

As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. Let t be the number of seconds that have elapsed since the Ferris wheel started. You find that it takes you 3 seconds to reach the top, 43 ft above the ground, and that the wheel makes a revolution once every 8 seconds. The diameter of the wheel is 40 ft. What is the value of t the second time you are 18 ft above the ground?

In this problem y is distance and x is time. First sketch a graph of what we know. After 3 seconds the distance is 43, and we know, since the period is 8 seconds, that the Ferris wheel will reach that distance again at 11 seconds. The third point of our sinusoidal graph can be found by adding 4 (half the period) to 3 to equal 7. At 7 seconds the Ferris wheel will be at the bottom of its rotation, which is 3 ft above the ground.

The vertical shift, or sinusoidal axis, of our equation is found by averaging the highest y value (43) and the lowest y value (3), which equals 23. The amplitude is found by subtracting the sinusoidal axis (23) from the highest y value (43), which equals 20. We already found the period to be 8, and the phase shift is 3 to the right because our highest y value is reached after 3 seconds.

So our cosine equation from our word problem is...



Now we can equal this equation to 18ft and simplify to begin to find the value of t the second time the Ferris wheel reaches 18ft. The simplified equation is...



Now we can use the following equations to find the appropriate x values





The calculator answer is as follows






The first equation can be set up and simplified like this...





The second equation can be set up and simplified like this...





We can solve this problem by finding the second positive x value for a y value of 18ft by plugging in integers for n. The first positive x value can be found by plugging in -1 for n in the second equation. This comes out to be .6783. The second positive x value, which is the value the problem is asking us for (when Ferris wheel will reach 18 feet for the second time), can be found by plugging in 0 for n in the first equation. This equals 5.3218 seconds and is the final answer to our problem.

Check out Mr Corn's website HERE for more information and for today's smart board slides.