Problem Statement:
The center of the high ferris roller in Las Vegas is 65 feet off of the ground, and has a radius of 50 feet. It takes 40 seconds for it to make a full revolution, starting at the 3 o’clock position. A diver wants to jump off a platform on the edge of the ferris wheel in time to land inside of a tub that is moving horizontally at 15 feet per second
How many seconds after the start of the ferris wheel revolution and the movement of the tub should the diver jump in order to successfully fall into the tub?
My class found this equation in order to find the solution to this problem.
-240+15[W(57+50sin9W16)]=50cos9W
This equation was constructed from a series of other factors that we took into consideration. Here is the breakdown:
Height
We used the following equation to find the height of the diving platform at any given time.
height = 65+50sin9t
The center of the ferris wheel is 65 feet off of the ground. Because it takes the ferris wheel 40 seconds to make a full revolution of 360 degrees, we concluded that it moves 9 degrees every second. So, with any amount of seconds, we can find the exact degree at which the platform lies. Then, we find the sine of that value, which, when multiplied by the 50 foot length of the radius, will give us the platform’s height off of the floor.
Horizontal Position
We also need to think about the horizontal position of the diver’s platform. To do this, we use the following equation.
position= 50cos(9t)
Again, using 9t we can find the degrees of the platform after a given time interval. Instead of using sine, we can use cosine to find the horizontal measurement of the same angle. Multiplying it by 50 will give us the position in relation to the size of the radius of the ferris wheel.
Tub
Now that we can find the position of the diver, we need to take into consideration the position of the tub. It is moving at a rate of 15 feet per second, starting at a position of -240. The tub is 8 feet tall and 10 feet wide.
Free Falling
Now that we know the position of the diver on both the vertical and horizontal axes of the ferris wheel, we must find the amount of time that it will take the diver to reach the tub. Gravity causes objects to fall at -16ft/sec. The equation is height/16.
Final Breakdown:
-240+15[W(57+50sin9W16)]=50cos9W
-240 is the starting position of the tub. The tub moves at the rate of 15 multiplied by W(57+50sin9W16) which is the amount of time it takes for the diver to reach the tub after a given time. 57+50sin9W gives the distance that the diver falls before hitting the tub
(Height from center of ferris wheel - Height of tub= 65 feet - 8 feet= 57 feet). This value is divided by 16 and square rooted to find the amount of time that it takes for them to fall. Then, that number is multiplied by W, the amount of time from the start of the rotation. This entire side of the equation is equal to the horizontal position of the diver at the given time, 50cos9W.
Personal Reflection
Through the process of working on this problem in class, my favorite and most commonly used mathematical quality that I used was explaining/justifying. This problem is a combination of many different components of a ferris wheel and includes algebra, trigonometry, and calculus. Having to work with these three concepts allowed me to expand my mind and my ways of thinking by ensuring that each step was bringing me closer to a final solution. Many times, I had to take a step back and understand why we were approaching a certain section of this problem, then explain it to my classmates through justification, deepening my own comprehension.
The center of the high ferris roller in Las Vegas is 65 feet off of the ground, and has a radius of 50 feet. It takes 40 seconds for it to make a full revolution, starting at the 3 o’clock position. A diver wants to jump off a platform on the edge of the ferris wheel in time to land inside of a tub that is moving horizontally at 15 feet per second
How many seconds after the start of the ferris wheel revolution and the movement of the tub should the diver jump in order to successfully fall into the tub?
My class found this equation in order to find the solution to this problem.
-240+15[W(57+50sin9W16)]=50cos9W
This equation was constructed from a series of other factors that we took into consideration. Here is the breakdown:
Height
We used the following equation to find the height of the diving platform at any given time.
height = 65+50sin9t
The center of the ferris wheel is 65 feet off of the ground. Because it takes the ferris wheel 40 seconds to make a full revolution of 360 degrees, we concluded that it moves 9 degrees every second. So, with any amount of seconds, we can find the exact degree at which the platform lies. Then, we find the sine of that value, which, when multiplied by the 50 foot length of the radius, will give us the platform’s height off of the floor.
Horizontal Position
We also need to think about the horizontal position of the diver’s platform. To do this, we use the following equation.
position= 50cos(9t)
Again, using 9t we can find the degrees of the platform after a given time interval. Instead of using sine, we can use cosine to find the horizontal measurement of the same angle. Multiplying it by 50 will give us the position in relation to the size of the radius of the ferris wheel.
Tub
Now that we can find the position of the diver, we need to take into consideration the position of the tub. It is moving at a rate of 15 feet per second, starting at a position of -240. The tub is 8 feet tall and 10 feet wide.
Free Falling
Now that we know the position of the diver on both the vertical and horizontal axes of the ferris wheel, we must find the amount of time that it will take the diver to reach the tub. Gravity causes objects to fall at -16ft/sec. The equation is height/16.
Final Breakdown:
-240+15[W(57+50sin9W16)]=50cos9W
-240 is the starting position of the tub. The tub moves at the rate of 15 multiplied by W(57+50sin9W16) which is the amount of time it takes for the diver to reach the tub after a given time. 57+50sin9W gives the distance that the diver falls before hitting the tub
(Height from center of ferris wheel - Height of tub= 65 feet - 8 feet= 57 feet). This value is divided by 16 and square rooted to find the amount of time that it takes for them to fall. Then, that number is multiplied by W, the amount of time from the start of the rotation. This entire side of the equation is equal to the horizontal position of the diver at the given time, 50cos9W.
Personal Reflection
Through the process of working on this problem in class, my favorite and most commonly used mathematical quality that I used was explaining/justifying. This problem is a combination of many different components of a ferris wheel and includes algebra, trigonometry, and calculus. Having to work with these three concepts allowed me to expand my mind and my ways of thinking by ensuring that each step was bringing me closer to a final solution. Many times, I had to take a step back and understand why we were approaching a certain section of this problem, then explain it to my classmates through justification, deepening my own comprehension.