A ferris wheel is 35 meters in diameter,and can be boarded at ground level. The wheel turns in a counterclokcwise direction, completing one full revolution every 5 minutes. Suppose that a t=0 you are in the three o'clock position. Write a formula, using the sine function for your height above ground after t minutes on the ferris wheel.
f = 1 REV / 5 min = 0.2 REV / min =
0.2 RPM = 0.2 Cycles / min. =Frequency.
Y = 35 Sin(360*f*t).
The amplitude (Y) should be max (35) at 1.25 minutes. Let's check it out!
Y = 35 * Sin (360*0.2*1.25) = 35 meters
Some sources use 2pi Radians instead of
360 deg.
To determine the formula for your height above ground after t minutes on the Ferris wheel, we can start by visualizing the situation.
First, we need to determine the radius of the Ferris wheel. The diameter is given as 35 meters, so the radius would be half of that, which is 35/2 = 17.5 meters.
Next, we can imagine the Ferris wheel as a circle, with you as a point on the circumference. At time t=0, you are located at the three o'clock position, which is horizontally aligned with the center of the circle.
As time progresses, the Ferris wheel turns counterclockwise, completing one full revolution every 5 minutes. This means that after t minutes, the wheel has turned (t/5) revolutions.
To determine your height above the ground, we can observe that as the wheel turns, your point on the circumference moves in a circular motion. We can project this motion onto the y-axis, where the top of the circle is the maximum height above ground and the bottom is the lowest.
The height above ground can be represented by the y-coordinate of your point on the circumference of the circle. To find this, we need to consider the relationship between the angle made by your position on the wheel and the height above the ground.
Since the wheel completes one full revolution in 5 minutes, we can use the fact that there are 12 positions on a clock face (3, 6, 9, etc.) to divide 360 degrees by 12, giving us 30 degrees per position.
Now, let's determine the angle made by your position on the wheel at time t. At t=0, you are at the three o'clock position, which corresponds to an angle of 0 degrees. As the wheel turns counterclockwise, the angle increases.
For every complete revolution the wheel makes (360 degrees), 5 minutes pass. Hence, the rate of change of the angle with respect to time is (360 degrees / 5 minutes) = 72 degrees per minute.
Therefore, at time t, the angle made by your position on the wheel can be calculated using the equation:
θ = 72t
Finally, to find your height above ground, we can use the sine function, which relates the angle of a right triangle to the ratio of the opposite side (height) over the hypotenuse (radius).
The formula for your height above ground after t minutes on the Ferris wheel, using the sine function, is:
h(t) = sin(θ) * r
Substituting the value of θ we obtained earlier:
h(t) = sin(72t) * r
Now, you can use this formula to calculate your height above ground for any given time t after the Ferris wheel starts turning.