Shirley is on a Ferris wheel which spins at the rate of 3.6 revolutions per minute. The wheel has a radius of 35 feet, and the center of the wheel is 51 feet above the ground. After the wheel starts moving, Shirley takes 12 seconds to reach the top of the wheel.
How high above the ground is she when the wheel has been moving for 8 minutes?
I know the formula is y(t)=rsin(wt+theta0)+ y0
I got the formula y(t) = 35sin(3.6pi/30 * t + (pi/2 - 3.6pi/30 * 12)) + 51 but when i plugged in 8 I got 53.19766818 but that's not right.. Is my formula off?
since the period is 60/3.6 seconds, and reaches a max at t=12 seconds,
y = 35cos(2pi/(60/3.6)(t-12))+51
which agrees with your formula. You have found y at t=8 seconds, not 8 minutes.
Your formula appears to be correct. However, it seems there was a mistake in the conversion of the time.
Given that the wheel spins at a rate of 3.6 revolutions per minute, we need to convert 8 minutes into seconds. Multiplying 8 by 60 gives us 480 seconds.
Using the formula y(t) = 35sin(3.6π/30 * t + (π/2 - 3.6π/30 * 12)) + 51, we can now solve for the height when the wheel has been moving for 8 minutes (480 seconds).
y(480) = 35sin(3.6π/30 * 480 + (π/2 - 3.6π/30 * 12)) + 51
Evaluating this expression, we find that Shirley is approximately 86.07 feet above the ground when the wheel has been moving for 8 minutes.
To find the height above the ground when the wheel has been moving for 8 minutes, we can use the formula you mentioned:
y(t) = r * sin(wt + theta0) + y0
Let's break down the formula:
- r is the radius of the wheel = 35 feet
- w is the angular velocity, which is given as 3.6 revolutions per minute. Since 1 revolution is equal to 2π radians, we can convert it to radians per minute by multiplying by 2π. So, w = (3.6 rev/min) * (2π rad/rev) = 7.2π rad/min.
- t represents the time in minutes. Since we want to find the height after 8 minutes, we'll substitute t = 8 into the formula.
- theta0 is the phase angle. It is the initial position of the rider at t = 0. To find theta0, we need to calculate the angle at which Shirley reaches the top of the wheel. We are given that it takes her 12 seconds (or 0.2 minutes) to reach the top. In radians, this would be (0.2 min) * (7.2π rad/min) = 1.44π rad. Since the wheel starts at 51 feet above the ground, this part of the formula becomes 51.
Now we can substitute these values into the formula:
y(8) = 35 * sin((7.2π * 8) + 1.44π) + 51
Calculating this expression:
y(8) = 35 * sin(57.6π + 1.44π) + 51
y(8) = 35 * sin(59.04π) + 51
y(8) ≈ 35 * sin(186.008) + 51
Using a calculator:
y(8) ≈ 35 * (-0.475) + 51
y(8) ≈ -16.625 + 51
y(8) ≈ 34.375
Therefore, when the wheel has been moving for 8 minutes, Shirley is approximately 34.375 feet above the ground.