Question

The Ferris wheelThe community loves the Ferris Wheel because when they are at the top of the wheel they can look for their house and it has long been a family favorite. The company sent you a Ferris wheel that has a radius of 40 ft. and each car is 5 ft. tall. The people can’t start to look for their houses until they get above the tents of the carnival and those tents are 20ft tall. The Ferris wheel has 20 passenger cars and it takes 12 minutes to have all the cars filled. Once all cars are filled the Ferris wheel runs at 1 rotation for every 2 minutes for a total of 8 minutes.Once someone has boarded the Ferris wheel, how long will it take for them to be able to start looking for their house?How long will they have to search for their house?

Answer 1

Given:

radius of Ferris wheel = 40 ft

height of a car = 5 ft

height of a tent = 20 ft

no. of passenger cars = 20

time to load all cars = 12 minutes

time for 1 rotation = 2 minutes

time for all rotations = 8 minutes

What we want to know is how many degrees the car will have to travel along the circle before it reaches a height of 20 ft.

We can draw a triangle to represent the unknown and find the angle.

The total height from the ground to the center of the Ferris wheel is 45 ft (5ft car height + 40 ft radius). We deduct the 20ft height of the tent so we get the side of the right triangle we're using to solve for the missing angle. That height is 25 ft.

The radius also serves as the hypotenuse of the triangle.

We can then use the following to solve for the angle theta:

`\begin{gathered} \cos\theta=(adjacent)/(hypotenuse) \\ \\ \cos\theta=(25)/(40) \\ \\ \cos^(-1)(\cos\theta)=\cos^(-1)((25)/(40)) \\ \\ \theta=51.3\degree \end{gathered}`

Now we know that the car must travel 51.3 degrees along the circle before it gets to a height where it can see above the tents. If one rotation is 360 degrees and it takes 2 minutes to complete one round, then to find the time it will take to travel 51.3 degrees, we use the following equation:

`\begin{gathered} (360\degree)/(2minutes)=\frac{51.3\degree}{x\text{ }minutes} \\ \\ x=(51.3(2))/(360) \\ \\ x=0.285\text{ minutes or }17.1\text{ seconds} \end{gathered}`

It will take them 17.1 to be able to start looking for their house.

Now every time the car goes around the path of the Ferris wheel, for 17.1 going up and another 17.1 seconds going down, they will not be able to look for their house.

17.1 seconds x 2 x 4 turns = 136.8 seconds

8 minutes - 136.8 seconds = 480 - 136.8 = 343.2 seconds or 5 minutes and 43.2 seconds

They have 5 minutes and 43.2 seconds to look for their house.