a ferris wheel 15m in diameter makes 1 revolution every 2 minutes. If the center of the wheel is 9m above the ground, how fast is the passenger in the wheel moving vertically when he is 12.5m above the ground?

20.84m/min

Well, let's break it down, shall we? The ferris wheel makes 1 revolution every 2 minutes, which means it completes 360 degrees in 2 minutes. Now, if we divide that by 2 (because we want to find out the speed at a specific point), we get 180 degrees in 1 minute.

Since the diameter of the wheel is 15m, we can find that the radius is half of that, which is 7.5m. So, when the passenger is 12.5m above the ground, he's actually at the top-most point of the ferris wheel, which is 9m above the ground. In other words, he's just hanging there enjoying the view!

Now, when the passenger is at this highest point, he's not moving vertically at all. So, the speed of the passenger in the wheel when he's 12.5m above the ground is... drumroll, please... zero! No need to worry about the speed when you're at the top, my friend. Just sit back, relax, and enjoy the ride... or lack thereof! 😄

To find the vertical speed of the passenger, we can use the concept of angular velocity.

The angular velocity (ω) of the ferris wheel can be calculated using the formula:
ω = (2π)/T

Where T is the time taken for one revolution. In this case, T = 2 minutes.

ω = (2π)/(2 minutes)
ω = π/minute

Next, we can find the linear velocity (v) of a point on the wheel using the formula:
v = ω * r

Where r is the radius of the wheel. In this case, r = 15m/2 = 7.5m.

v = (Ï€/minute) * (7.5m)
v = 7.5Ï€ m/minute

Now, we can find the rate of change of height (dv/dt) of the passenger when he is 12.5m above the ground.

Since the passenger is moving vertically, the rate of change of height (dv/dt) will be equal to the vertical velocity of the passenger.

Therefore, the vertical velocity of the passenger when he is 12.5m above the ground is equal to:
v_vertical = dv/dt

To find this, we can use similar triangles. The distance from the ground to the passenger (d) is given by:
d = 9m + 12.5m = 21.5m

Since the ferris wheel is rotating and the passenger is moving in a circle, we can form a right triangle with the radius of the wheel (7.5m), the distance from the ground to the passenger (21.5m), and the distance from the center of the wheel to the passenger (h).

Using the Pythagorean theorem, we can find the value of h:
h^2 = 21.5m^2 - 7.5m^2
h^2 = 462.5m^2
h = √462.5m
h = 21.5m

Since the passenger is moving in a circle, the vertical velocity (v_vertical) can be found by differentiating the height (h) with respect to time (t), using the chain rule:
v_vertical = dh/dt

So, we need to find dh/dt when h = 21.5m.

Let's solve for dv/dt using the chain rule.

Using the Pythagorean theorem, we have:
21.5m^2 = r^2 + h^2
21.5m^2 = (7.5m)^2 + h^2
h^2 = 21.5m^2 - (7.5m)^2
h^2 = 262.5m^2
h = √262.5m
h = 16.18m

Now, let's differentiate both sides of the equation with respect to t:
2h * (dh/dt) = 0

Substituting the known values, we have:
(2 * 16.18m) * (dh/dt) = 0
32.36m * (dh/dt) = 0
32.36m * (dv/dt) = 0

Therefore, the rate of change of height (dv/dt) when the passenger is 12.5m above the ground is 0.

To find the vertical speed of the passenger, we can use the concept of angular velocity and trigonometry.

First, let's determine the angular velocity of the Ferris wheel.

The Ferris wheel makes one revolution every 2 minutes, which means it completes 360 degrees in 2 minutes. Therefore, the angular velocity can be calculated as:

Angular velocity = 360 degrees / 2 minutes
Angular velocity = 180 degrees/minute

Now, let's consider the position of the passenger.

When the passenger is 12.5m above the ground, we can create a right-angled triangle with one side representing the distance between the passenger and the center of the Ferris wheel, and the other side representing the vertical distance of the passenger from the ground.

The diameter of the Ferris wheel is 15m, so the radius (distance between the center and the passenger) is half of that, which is 7.5m.

Using Pythagorean theorem, we can find the other side of the triangle, which represents the horizontal distance between the passenger and the center:

Horizontal distance = sqrt((15m)^2 - (7.5m)^2)

Now that we have the horizontal and vertical distances, we can differentiate them with respect to time to find the rates of change.

Differentiating the horizontal distance with respect to time gives us the horizontal speed, which will be zero since the horizontal distance does not change.

Differentiating the vertical distance with respect to time gives us the vertical speed.

Now, let's substitute the values into the equation:

Vertical speed = (Vertical distance / Horizontal distance) * Angular velocity

Vertical distance = 12.5m
Horizontal distance = sqrt((15m)^2 - (7.5m)^2)
Angular velocity = 180 degrees/minute

After substituting the known values, we can evaluate the equation to find the vertical speed of the passenger when he is 12.5m above the ground.