When a wheel is rotated through an angle of 35 degrees, a point on the circumference travels through an arc length of 2.5 m. When the wheel is rotated through angles of 35 rad and 35 rev, the same point travels through arc lengths of 143 m and 9.0 X 10^2 m, respectively. What is the radius of the wheel?

Question

When a wheel is rotated through an angle of 35 degrees, a point on the circumference travels through an arc length of 2.5 m. When the wheel is rotated through angles of 35 rad and 35 rev, the same point travels through arc lengths of 143 m and 9.0 X 10^2 m, respectively. What is the radius of the wheel?

I know that the radius is the change in arc length divided by the change in the angle in radians, but I don't understand the second sentence (the 35 rev part).

Answer 1

The first sentence is all you need.

arc distance=radius*angleinRadians
2.5=radius(35deg*PIrad/180deg)
radius=4.09 m

Answer 2

I think the first sentence is extra information. Here's my work:

Change in arc/change in radians = 757 / 184.9 = 4.09 meters

To get the change in arc, I did:

900m - 143m = 757 m

To get the change in radians, I did:

70pi radians - 35 radians = 184.9 radians

Is my answer of 4.09 m for the radius correct?

Answer 3

thanks! :D

Answer 4

thanks! :D

though some more explanation would be appreciated for example why (35*PIrad/180deg), because you are converting degrees into radians

Answer 5

Ah, the mysterious world of rotations! Let me enlighten you with a touch of humor.

Sounds like that wheel is having quite the journey! So, let's break it down.

When the wheel is rotated 35 degrees, the point on the circumference travels through an arc length of 2.5 m.

Now, when the wheel is spun around even further, let's say 35 radians, that same point on the circumference journeys through a much longer arc length of 143 m.

But what's this? The wheel wants to show off even more! When it's rotated 35 revolutions (or revs, as we hip mathematicians like to call them), the point on the circumference takes a wild, wacky ride along an even longer arc length of 9.0 x 10^2 m.

Now, to find the radius of the wheel, we can use the formula you mentioned: radius = change in arc length / change in angle in radians.

For the second part of your question, where the wheel rotates 35 revs, we need to convert those revs into radians.

Since 1 revolution is equal to 2π radians, we can multiply 35 revs by 2π to get the change in angle in radians.

So, the change in angle in radians is 35 revs * 2π = 70π radians.

Now we can calculate the radius using the formula!

For the first scenario: radius = 2.5 m / 35 degrees.

For the second scenario: radius = 143 m / (70π radians).

And for the third, adventurous scenario: radius = 9.0 x 10^2 m / (70π radians).

Plug in these numbers and you'll find the radius that's been taking the point on the wheel's thrilling ride through different arc lengths.

Just remember, when it comes to wheels, angles, and revolutions, things can get a little dizzying. But with a dash of humor, we can unravel the mathematical madness together!

Answer 6

In order to understand the second sentence, we need to clarify the concept of radians and revolutions.

- Radians (rad) are a unit of measurement for angles. One radian is equal to the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. In other words, the circumference of a circle is equal to 2π radians.
- Revolutions (rev) are a unit of measurement for angles as well. One revolution is equal to a complete trip around the circumference of a circle, which is equal to 2π radians.

Now, let's break down the given information:

1. When the wheel is rotated through an angle of 35 degrees, the point on the circumference travels through an arc length of 2.5 m. This means that the change in angle is 35 degrees or approximately 0.61 radians (using the conversion factor: 180 degrees = π radians). The change in arc length is given as 2.5 m.

2. When the wheel is rotated through an angle of 35 radians, the point on the circumference travels through an arc length of 143 m. This means that the change in angle is 35 radians (as given). The change in arc length is given as 143 m.

3. When the wheel is rotated through an angle of 35 revolutions, the point on the circumference travels through an arc length of 9.0 x 10^2 m. This means that the change in angle is 35 revolutions, which is equivalent to 70π radians (since one revolution is 2π radians). The change in arc length is given as 9.0 x 10^2 m.

Now, let's calculate the radius of the wheel using the formula you mentioned:

radius = change in arc length / change in angle

For the first case:
radius = 2.5 m / 0.61 rad ≈ 4.1 m

For the second case:
radius = 143 m / 35 rad ≈ 4.1 m

For the third case:
radius = 9.0 x 10^2 m / (70π rad) ≈ 4.06 m

So, based on the given information, the radius of the wheel is approximately 4.1 meters.