1. At the amusement park, you decide to ride the Ferris wheel, which has a maximum height of 50 meters and a diameter of 35 meters. It takes the wheel three minutes to make one revolution. If you start your ride at the midline and the ferris wheel rotates counter clockwise, how many seconds will it take for your seat to reach a height of 50 meters? Round the answer to the nearest second.
a.)59 seconds
b.)58 seconds
c.)45 seconds
d.)15 seconds
2. At the amusement park, you decide to ride the Ferris wheel, which has a maxmum height of 80 meters and a diameter of 40 meters. It takes the wheel seven minutes to make one revolution. If you start your ride at the midline and the ferris wheel rotates counter clockwise, how many seconds will it take for your seat to reach a height of 60 meters? Round the answer to the nearest second.
a.)0 seconds
b.)10 seconds
c.)20 seconds
d.)40 seconds
Thanks!!!
To solve both of these problems, we need to calculate the angular velocity of the Ferris wheel and then determine the time it takes for your seat to reach the desired height.
1. Let's start with the first problem. The Ferris wheel has a diameter of 35 meters, so its radius is half of that, which is 17.5 meters. It takes 3 minutes for the wheel to make one revolution, which corresponds to a full circle of 360 degrees.
To find the angular velocity, we can use the formula: angular velocity (ω) = 360 degrees / time for one revolution.
So, ω = 360 degrees / 3 minutes.
Now, we want to find how long it takes for your seat to reach a height of 50 meters. Since the starting point is the midline, we need to calculate the angle between the midline and the position at the desired height.
The midline is the bottommost point of the Ferris wheel, which is located at a height of 0 meters. The desired height is 50 meters. The radius of the wheel is 17.5 meters.
Using trigonometry, we can find the angle θ between the midline and the desired height using the formula: sin(θ) = (opposite side) / (hypotenuse).
So in this case, sin(θ) = 50 meters / 17.5 meters.
Now that we know the angle θ, we can calculate the time it takes for your seat to reach the desired height using the formula: time = θ / ω.
2. For the second problem, the process is the same. The Ferris wheel has a diameter of 40 meters, so its radius is 20 meters. It takes 7 minutes for one revolution, corresponding to 360 degrees.
Using the same formulas as before, we can calculate the angular velocity (ω) and the angle (θ) between the midline and the desired height of 60 meters. Finally, we can determine the time it takes for your seat to reach the height of 60 meters using the formula: time = θ / ω.
By following these steps, you can solve both problems and find the correct answers.
1. To find out how long it will take for your seat to reach a height of 50 meters, we can use the concept of angular velocity.
First, let's find the circumference of the Ferris wheel using the given diameter:
Circumference = π * Diameter = π * 35 meters ≈ 110 meters
Since the Ferris wheel takes three minutes to make one revolution, we need to convert this to seconds:
Time for one revolution = 3 minutes * 60 seconds/minute = 180 seconds
Now, we can calculate the angular velocity, which is the angle covered in a given time period:
Angular velocity = 360 degrees / Time for one revolution = 360 degrees / 180 seconds = 2 degrees/second
To reach a height of 50 meters, your seat needs to move to the topmost position. At this point, the angle covered will be 180 degrees, as the topmost position is halfway around the circle (180 degrees from the starting position at the midline).
Now, we can calculate the time it takes for your seat to reach a height of 50 meters:
Time = Angle covered / Angular velocity = 180 degrees / 2 degrees/second = 90 seconds
Rounded to the nearest second, it will take approximately 90 seconds for your seat to reach a height of 50 meters.
Therefore, the correct answer is not provided in the given options.
2. Similar to the previous question, we can use the concept of angular velocity to find the time it takes for your seat to reach a height of 60 meters.
Using the given diameter, the circumference of the Ferris wheel is:
Circumference = π * Diameter = π * 40 meters ≈ 125.66 meters
The Ferris wheel takes seven minutes to make one revolution, which is equal to 420 seconds.
Calculating the angular velocity:
Angular velocity = 360 degrees / Time for one revolution = 360 degrees / 420 seconds ≈ 0.857 degrees/second
When your seat is at a height of 60 meters, it has covered an angle of:
Angle covered = 360 degrees * (60 meters / Circumference) = 360 degrees * (60 meters / 125.66 meters) ≈ 172.7 degrees
The time it takes for your seat to reach a height of 60 meters can be calculated as:
Time = Angle covered / Angular velocity ≈ 172.7 degrees / 0.857 degrees/second ≈ 201.74 seconds
Rounded to the nearest second, it will take approximately 202 seconds for your seat to reach a height of 60 meters.
Therefore, the correct answer is not provided in the given options.
no ideas of your own to provide? These two problems are really the same thing with different numbers. As you might imagine, the trick is to come up with a function you can analyze.
So, on the first one,
if we are starting at the midline, then since sin(0) = 0, we expect to have some variation on
y = sin(x)+
diameter of 35m means a radius of 17.5m So, you will have
y = 17.5 sin(x)
The midline is one radius velow the maximum, so
y = 50-17.5sin(x)
The period is 3 minutes, so if t is in minutes,
y = 50 - 17.5 sin(2π/3 x)
Now you can work out the questions about what happens when.
Follow the same steps for the 2nd one.
Actually, the first one is so easy we wasted our time building up a function. Start-to-top is 1/4 revolution, or 3/4 minutes.