A Ferris wheel is 10 meters in diameter and boarded from a platform that is 3 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 2 minutes. How many minutes of the ride are spent higher than 10 meters above the ground?
Follow the same steps that I used in your next question.
Let me know what you get.
To find out how many minutes of the ride are spent higher than 10 meters above the ground, we need to determine the height of the Ferris wheel at each point in time during its rotation.
First, let's calculate the radius of the Ferris wheel. The diameter is given as 10 meters, so the radius (r) is half of the diameter, which is 10/2 = 5 meters.
Next, we need to determine the height of the Ferris wheel at the six o'clock position, which is level with the loading platform. Since the platform is 3 meters above the ground, the height at the six o'clock position is equal to the radius plus the platform height: 5 + 3 = 8 meters.
Now, let's consider the highest point of the Ferris wheel. As it completes one full revolution, the highest point will be at the top of the circle, which is at a distance of two times the radius from the center. So, the highest point is at a height of 2 * 5 = 10 meters.
Since we want to find the minutes spent higher than 10 meters, we need to find the time during the ride when the height is greater than 10 meters.
During a complete revolution of the Ferris wheel, there are 2 * π radians, where π (pi) is a mathematical constant approximately equal to 3.14159. Since the Ferris wheel completes one full revolution in 2 minutes, the angular velocity is 2 * π / 2 = π radians per minute.
Now, let's set up an equation to determine the time spent higher than 10 meters. Let t represent the time in minutes, and h(t) represent the height at time t.
We know that h(t) = r * cos(π * t), where r is the radius of the Ferris wheel and π is the angular velocity.
To find the time spent higher than 10 meters, we need to solve the following inequality: h(t) > 10.
r * cos(π * t) > 10
5 * cos(π * t) > 10
cos(π * t) > 10/5
cos(π * t) > 2
Now, to solve this inequality, we can inspect the values of cos(π * t) as t increases.
The cosine function oscillates between -1 and 1. Since we know that the Ferris wheel is never below 8 meters (at the six o'clock position), the height cannot be greater than 10 meters for any value of t. Therefore, there are no minutes during the ride spent higher than 10 meters above the ground.