Question 1182937: A Ferris Wheel having a diameter of 100 feet and a maximum height of 110 feet rotates counter clockwise through an angle of 1.5 revolutions per minute. Note that 1 revolution is equivalent to 360°
a. If your starting location is 50 feet East of the Ferris Wheel’s center and 60 feet above the ground, what will be your approximate location above the ground after 8 seconds?
b. At a given time, you are 40 feet above the ground. What is the least amount of time it takes for you to be on the same exact location?
c. Which approximate angle measure, in degree, is being formed by the horizontal midline of the Ferris Wheel and your current position if you are 100 feet above the ground and 30 feet west of the vertical midline of the Ferris Wheel?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the ferris wheel makes 1.5 revolutions per minute.
translate that to seconds and it makes 1.5 revolutions every 60 seconds.
divide 60 by 1.5 and you see that it makes one revolution every 40 seconds.
the center point of the circle formed is 60 feet above the ground.
the radius of the circle formed is 50 feet.
the starting location is 50 feet east of the ferris wheel's center and 60 feet above the ground.
the bottom of the circle is 10 feet above the ground.
the top of the circle is 110 feet above the ground.
the starting location is 50 feet east of the center of the circle.
that makes it 60 feet above the ground.
since 1 revolution is 360 degrees in 40 seconds, then the location of any point on the circle moves 9 degrees every second.
your questions:
a. If your starting location is 50 feet East of the Ferris Wheel’s center and 60 feet above the ground, what will be your approximate location above the ground after 8 seconds?
the approximate location after 8 seconds will be 107.553 feet above the ground.
b. At a given time, you are 40 feet above the ground. What is the least amount of time it takes for you to be on the same exact location?
the least amount of time to get to 40 feet above the ground again appears to be 14.76 seconds.
c. Which approximate angle measure, in degree, is being formed by the horizontal midline of the Ferris Wheel and your current position if you are 100 feet above the ground and 30 feet west of the vertical midline of the Ferris Wheel?
the approximate angle measure, in degrees, is 126.8098976 degrees.
divide that by 9 and it corresponds to 14.09665529 seconds after starting.
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these are your answers.
getting them was a little difficult.
i was able to graph the equation to get / check my answers.
i used the unit circle to visually model the ferris wheel.
here's my graph in seconds.
here's my graph in degrees.
the equation in seconds is 50 * sin(9x) + 60.
the equation in degrees is 50 * sin(x) + 60.
the equation in seconds changed the frequency from 1 to 9.
frequency = 360 / period.
with a period of 40, the frequency becomes 360 / 40 = 9.
the general form of the equation used is y = a * sin(b * (x - c)) + d
a is the amplitude which was 50
b is tyhe frequency which ws 9 for seconds and 1 for degrees.
c is the horizontal displacement which was 0.
d is the vertical displacement which was 60.
the vertical displacement is the horizontal center line of the graph.
with a horizontel center line of 60, the highest amplitude would be 60 + 50 = 110 and the lowest amplitude would be 60 - 50 = 10.
here is a picture of the starting location on the ferris wheel.
that would be point A.
here is a picture of the approximate location after 8 seconds.
here is a picture when the location on the ferris wheel is 40 feet above the ground.
here is a picture of when the location is 100 feet above the ground and 30 feet to the west of the vertical center line.
there's a lot here, making it difficult to fully answer the methods used to get the answer.
if you have any questions, send me a note and i'll reply as soon as i can.
Answer by ikleyn(52798)  (Show Source):
You can put this solution on YOUR website! .
I'd like to make a correction to the answer to part (b), provided by @Theo.
The question asks
What is the least amount of time it takes
for you to be on the same exact location?
It does not ask about getting the same height - it asks about getting "the same exact location".
Therefore, the right answer, in my opinion, is the time for one full exact rotation, i.e. 40 seconds.
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