Visualize a scene in an amusement park. Picture a bright day with a clear blue sky. In the center, construct a playful merry-go-round, intricately designed, with a radius of 25 feet, spinning at a slow but constant pace. Upon this device, imagine a young woman, dressed casually in bright, summer colors. She seems excited and is preparing to jump off. In the crowd around the merry-go-round, show a collection of individuals, representing a diverse array of ages and backgrounds, caught in various states of enjoyment, surprise, and anticipation. Do not include any text in the image.

Mary is riding a merry-go-round, whose radius is 25 feet and which is turning 36 degrees per second. Seeing a friend in the crowd, she steps off the the outer edge of the merry-go-round and suddenly finds it necessary to run? At how many miles per hour does she have to run?

At the moment she steps off, her linear velocity equals the rate the arc length is increasing.

circumference = 2π(25) = 50π ft
36° = 36/360 = 1/10 of a rotation

so the linear velocity of any point is 5π ft/second

5π ft/1 second
= (5π/5280) / (1/3600) miles/hour
= (5π/(5280)(3600) mph
= appr 10.7 mph

since 1 revolution is 360 degrees,

(25 * 2π ft/rev) * (2π/10 rev/s) * 3600s/hr * 1mi/5280 ft = 10.71 mi/hr

Well, Mary certainly has some running to do! Let's calculate that for her.

First, we need to convert the radius from feet to miles. One mile is equal to 5280 feet, so the radius is 25/5280 = 0.0047348 miles.
Now, we need to figure out how fast Mary needs to run. Since the merry-go-round is turning at a rate of 36 degrees per second, Mary needs to match that speed to stay in one place on the edge.
To calculate that, we need to find the circumference of the merry-go-round using 2πr = 2 × π × 0.0047348 = 0.0297788 miles.
Since she has to run the same distance in the same time, Mary has to run 0.0297788 miles per second.
To convert that to miles per hour, we multiply by the number of seconds in an hour: 0.0297788 × 3600 = 107.20368 miles per hour.
So, Mary needs to run at approximately 107.20 miles per hour! That's faster than any superhero I know!

To determine how fast Mary needs to run, we need to calculate the speed at which she steps off the merry-go-round.

First, let's convert the angle from degrees to radians since trigonometric functions use radians.
36 degrees * (π/180) = 0.6283 radians

The linear velocity of an object moving in a circular path can be calculated using the formula:
v = r * ω
where v is the linear velocity, r is the radius, and ω is the angular velocity.

Given that the radius of the merry-go-round is 25 feet and the angular velocity is 0.6283 radians/second:
v = 25 ft * 0.6283 rad/s
v ≈ 15.7075 ft/s

Now, we can convert the linear velocity from feet per second to miles per hour:
1 mile = 5280 feet
1 hour = 3600 seconds

v = 15.7075 ft/s * (1 mile/5280 ft) * (3600 s/1 hr)
v ≈ 10.6896 mph

Therefore, Mary needs to run at approximately 10.69 miles per hour.

To determine the speed at which Mary needs to run, we first need to calculate her initial linear velocity while she is on the merry-go-round.

The linear velocity (v) of an object moving in a circular path can be found using the formula:
v = r * ω,
where:
v is the linear velocity,
r is the radius of the circular path, and
ω is the angular velocity in radians per second.

Given that the radius (r) of the merry-go-round is 25 feet and the angular velocity (ω) is 36 degrees per second, we need to convert the angular velocity to radians before calculating the linear velocity.

1. Converting angular velocity from degrees to radians:
ω (in radians per second) = ω (in degrees per second) * (π/180).

Here, ω = 36 degrees/sec, so:
ω (in radians per second) = 36 * (π/180) = 0.628 radians per second (approximately).

Now, we can calculate the linear velocity (v) using the formula:
v = r * ω.

v = 25 feet * 0.628 radians per second = 15.7 feet per second (approximately).

Since Mary steps off the outer edge of the merry-go-round, she needs to run with a linear velocity equal to or greater than the initial linear velocity (v). Therefore, she needs to run at least 15.7 feet per second to match the speed of the merry-go-round.

To convert this linear velocity to miles per hour, we can use the following conversion factors:
1 mile = 5280 feet and 1 hour = 3600 seconds.

Converting feet per second to miles per hour:
15.7 feet per second * (1 mile / 5280 feet) * (3600 seconds / 1 hour) = 10.7 miles per hour (approximately).

Therefore, Mary needs to run at approximately 10.7 miles per hour to match the speed of the merry-go-round.