Shirley is on a ferris wheel which spins at the rate of 3.2 revolutions per minute. The wheel has a radius of 45 feet, and the center of the wheels is 59 feet above the ground. After the wheel starts moving. Shirley takes 16 seconds to reach the top of the wheel. How high above the ground is she when the wheel has been moving for 9 minutes?
From the information above this is what I did...
w=3.2 RPM and I changed it to 6.4 rad/min
Radius=45 ft
t= 16 secs and I changed it to 4/15 mins
and I'm suppose to plug it in into this formula right?
x(t)=rcos( θ+wt)
y(t)=rsin( θ+wt)
I need help finding theta
3.2 rev/min = 20.1 rad/min. You forgot the factor pi
If she is moving at this rate, and the rate is constant, it should take
(1/2)(1/3.2) = 1/6.4 minutes = 9.4 seconds to go from the bottom to the top of the wheel. Something is inconsistent with the numbers you have been given
To find the angle θ, we can use the information given that Shirley takes 16 seconds to reach the top of the wheel. This means that in 16 seconds, she completes one full revolution on the ferris wheel.
Since we know the ferris wheel spins at a rate of 3.2 revolutions per minute, we can calculate the number of seconds it takes for one revolution:
1 revolution = 60 seconds / 3.2 revolutions per minute = 18.75 seconds
Therefore, each revolution corresponds to an angle of 2π radians.
Now, we need to calculate the time it takes for Shirley to reach a certain point on the ferris wheel. In this case, we want to find the height above the ground after the wheel has been moving for 9 minutes.
We know that 1 minute is equal to 60 seconds, so 9 minutes would be equal to:
9 minutes * 60 seconds/minute = 540 seconds
Next, we'll convert this time to radians by using the rate of revolution:
540 seconds / 18.75 seconds/revolution = 28.8 revolutions
Since each revolution corresponds to an angle of 2π radians, we can calculate the angle θ as:
θ = 28.8 revolutions * 2π radians/revolution
Now that we have the angle θ, we can plug it into the formulas you provided to find the height above the ground.
x(t) = r * cos(θ + wt)
y(t) = r * sin(θ + wt)
In this case, we have:
r = 45 ft (radius)
w = 6.4 radians/minute (angular velocity)
t = 9 minutes
Plugging in these values, we can calculate the height above the ground using the formulas.
To find the value of θ, we need to determine the initial angle of Shirley when she starts her ride on the Ferris wheel.
Given that Shirley takes 16 seconds to reach the top of the wheel, we can calculate the angle θ using the formula:
θ = (t / T) * 2π
where t is the time it takes for Shirley to reach the top (16 seconds) and T is the period of the Ferris wheel, which can be calculated by dividing 1 minute (60 seconds) by the rate of revolution (3.2 revolutions per minute):
T = 60 seconds / 3.2 revolutions per minute
T = 18.75 seconds/revolution
Plugging in the values, we find:
θ = (16 seconds / 18.75 seconds/revolution) * 2π
θ = (16 / 18.75) * 2π
Now, let's calculate θ:
θ = (16 / 18.75) * 2π
θ ≈ 1.022 π
So, the initial angle θ is approximately 1.022π.
Now you can use the formulas for x(t) and y(t) to find how high above the ground Shirley is when the wheel has been moving for 9 minutes by replacing θ with 1.022π and plugging in the appropriate values for r (radius), w (angular velocity), and t (time).