A ferris wheel is 28 meters in diameter and makes one revolution every 10 minutes. For how many minutes of any revolution will your seat be above 21 meters?
fit your data into a sine curve
period of sine curve = 10 minutes
2π/k = 10
k = 2π/10 = π/5
so start with
y = 14 sin π/5 t + 14
testing:
t = 0: y = 14sin0 + 14 = 14
t = 2.5 , y = 28
t = 5 , y = 14
t = 7.5 , y = 0
t = 10 , y = 14 , looks good
solve for
14 sin πt/5 + 14 = 21
sin πt/5 = 1/2
πt/5 = π/6 or πt/5 = 5π/6
t = 5/6 or t = 25/6 minutes
so time over 21 metres = 25/6 - 5/6 = 20/6 minutes
or 2 minutes and 20 seconds or 200 seconds
I'm sure the answer is unchanged, but I'd have set up the equation as
y = 14(1 - cos pi/5 t)
That way, y(0) = 0, when the person gets on the wheel, rising to y(5) = 28 when it's 5 minutes later, at the top. And so on.
To solve this problem, we need to determine the portion of the revolution during which the seat is above 21 meters.
First, let's find the circumference of the ferris wheel. The circumference is calculated by multiplying the diameter by Pi (π), where Pi is approximately 3.14.
Circumference = Pi * Diameter
Circumference = 3.14 * 28 meters
Circumference = 87.92 meters (rounded to two decimal places)
Now, we need to determine the time it takes for the ferris wheel to complete one revolution, which is given as 10 minutes.
To find the portion of the revolution during which the seat is above 21 meters, we can use proportions:
Proportion: (portion of the revolution above 21 meters) / (time for one revolution) = (height above 21 meters) / (full circumference)
Let's plug in the values:
Proportion: (portion of the revolution above 21 meters) / 10 minutes = 21 meters / 87.92 meters
To solve for the proportion of the revolution, we can cross-multiply and solve:
(portion of the revolution above 21 meters) = (10 minutes * 21 meters) / 87.92 meters
(portion of the revolution above 21 meters) = 2.39 minutes (rounded to two decimal places)
Therefore, the seat will be above 21 meters for approximately 2.39 minutes during any given revolution.
To solve this problem, we need to understand how the position of the seat changes as the ferris wheel rotates.
The diameter of the ferris wheel is given as 28 meters, which means the radius is half of that, or 28/2 = 14 meters.
Since the ferris wheel makes one revolution every 10 minutes, we can infer that it completes 360 degrees of rotation in this time.
Now, let's find out how much of a revolution is needed for the seat to be above 21 meters.
First, we need to determine at what angle the seat is at when it's at 21 meters above the ground. To do this, we use trigonometry and the fact that we know the radius (14 meters).
The angle θ, measured in degrees, can be found using the equation:
cos(θ) = Adjacent/Hypotenuse
In this case, the "Adjacent" side is 21 meters (the height above the ground) and the "Hypotenuse" is 14 meters (the radius).
cos(θ) = 21/14
Using a calculator, we find that cos(θ) ≈ 1.50.
To find the value of θ, we can take the inverse cosine (cos^(-1)) of 1.50.
cos^(-1)(1.50) ≈ 55.07 degrees.
This means that the seat is above 21 meters for an angle of approximately 55.07 degrees.
Now, since we know that a full revolution is 360 degrees, we can calculate the proportion of a full revolution that corresponds to 55.07 degrees.
Proportion = Angle/360
= 55.07/360
≈ 0.1524
Therefore, the seat is above 21 meters for approximately 0.1524 of a full revolution.
To convert this fraction of a revolution to minutes, we multiply it by the total time for one revolution, which is 10 minutes.
Time = 0.1524 * 10
≈ 1.524 minutes
Therefore, the seat will be above 21 meters for approximately 1.524 minutes of any revolution.