Easy trig prob what did I do wrong?

[Miike012]

Homework Statement

Find number of deg. subtendedat the center of a circle by an arc whose length is 357 times the radius,taking pi = 3.1416...

The Attempt at a Solution

The arc length is... 357(r)
The radius length is r

One revolution = 2r(pi) = arc lenght.

Now I will solve...
2r(pi)x = 357(r)
x = 357/2(pi)

(360 deg = 2r(pi))(357/2(pi))

357(r) = 20454.5 deg.

In the back of the book it says...20.4545

As you can see I am off a couple decimals... what happened?

 

[Mentallic]

The book failed, that's what happened.

20^o is tiny. It's obviously wrong

 

[Miike012]

So are you saying I am correct?

 

[Mentallic]

Yes that's exactly what I'm saying.

By the way, just remember that $$\pi \equiv 180^o$$ so $$357 = 357\cdot\frac{\pi}{\pi} = 357\cdot \frac{180^o}{\pi}=20454^o$$

 

[Miike012]

It looks like you took the arc and mult by a rad and that gave you deg?
I thought you could only take rad and convert it to deg.

 

[Miike012]

What does the triple line between the pi and 180 sign mean?

 

[HallsofIvy]

The entire circumference is only $2\pi= 6.18...$ times the radius so an arc that is "357 times the radius" is far more than a single entire circle. Since the idea of an arc looping back on itself is peculiar, I suspect that the problem should have been to "find an angle so that the are is 0.357 times the radius". That would give
$$\frac{\theta}{180}= \frac{0.357}{\pi}$$
so that $\theta= 20.45$ degrees.

 

[Mentallic]

The entire circumference is only $2\pi= 6.18...$ times the radius so an arc that is "357 times the radius" is far more than a single entire circle. Since the idea of an arc looping back on itself is peculiar, I suspect that the problem should have been to "find an angle so that the are is 0.357 times the radius". That would give
$$\frac{\theta}{180}= \frac{0.357}{\pi}$$
so that $\theta= 20.45$ degrees.

Oh right, that might certainly be what it was looking for. I guess I misinterpreted the question just like the OP did.

What does the triple line between the pi and 180 sign mean?

It means they're equivalent. Just think of it as equals for the moment because it doesn't have many big differences.

It looks like you took the arc and mult by a rad and that gave you deg?
I thought you could only take rad and convert it to deg.

The arc is measures in radians and you can convert between radians and degrees freely.