Question

Tammy is at the dentist's office waiting on her appointment. She notices that the 6-inch-long minute hand is rotating around the clock and marking off time like degrees on a unit circle.

Part 1: How many radians does the minute hand move from 1:20 to 1:55? (Hint: Find the number of degrees per minute first.)
Part 2: How far does the tip of the minute hand travel during that time?

You must show all of your work.

Answer 1

@zepdrix

Answer 2

Were you able to solve the hint?
Degrees per minute = ?

Answer 3

No. I wasn't :(

Answer 4

was it 210?

Answer 5

A full rotation is 360 degrees.
A full rotation is also 60 minutes, yes?

So what is the ratio of degrees to minutes?\[\large\rm \frac{360}{60}=?\]

Answer 6

6

Answer 7

So there are 6 degrees in each minute. Ok great.

Answer 8

So as follow up questions I'll ask:
`How many minutes are there between 1:20 and 1:55?`
and then using the Hint,
`How many degrees are there between 1:20 and 1:55?`

Answer 9

hmm so i am assuming 210 wouldnt be the answer right?

Answer 10

oh okay sorrry! 35 mintues

Answer 11

Ok great so you've answered my second question.
210 degrees.

Remember how to convert to radians? :)

Answer 12

um is it 180 times x/pi?

Answer 13

is it 35 min. from 1:20-1:55

Answer 14

35 minutes from 1:20 to 1:55
6 degrees per minute,
35*6 degrees from 1:20 to 1:55.

Yes, good. Hopefully that part makes sense.

Answer 15

180 degrees and pi are equivalent on our unit circle.\[\large\rm 180^o=\pi\]We can either divide by pi to get,\[\large\rm \color{royalblue}{\frac{180^o}{\pi}=1}\]Or we can instead divide by 180 degrees to get,\[\large\rm \color{royalblue}{\frac{\pi}{180^o}=1}\]We want to use one of these blue relationships to deal with our 210 degrees.

Answer 16

Yay! Ok I see

Answer 17

You have 210 degrees.
Think of that value like this: \(\large\rm 210^o=\dfrac{210^o}{1}\)

So if our degrees are in the numerator,
which fraction should we use to cancel out the degrees and end up with radians?

Answer 18

Which blue one?

Answer 19

2nd one?

Answer 20

Ahh, well put detective!
\[\large\rm \frac{210^o}{1}\cdot\color{royalblue}{1}\quad=\frac{210^{o}}{1}\cdot\color{royalblue}{\frac{\pi}{180^{o}}}\]Degrees in the bottom will cancel with our degrees in the top,\[\large\rm \frac{210^{\cancel o}}{1}\cdot\frac{\pi}{180^{\cancel o}}\quad=\frac{210\pi}{180}\]and simplify.

Answer 21

hmm ok I'm bad at simplifying fractions, so I'll go with 7pi/12?

Answer 22

So taking the zero out is a nice first step.
21pi/18

Then I think we can take out.... a 3 from that point, ya?

Answer 23

yes:)

Answer 24

Hmm 18 divided by 3 is not 12. :d

Answer 25

it is 6

Answer 26

Oh I guess I made a little boo boo.
Yes, we have a radian measure of 7pi/6, good.
But that's for a circle of radius 1.

Notice that our minute hand is 6 inches, so our radius is 6.

Answer 27

ok so the answer to part 1 would be 7pi/6?

Answer 28

No, we'll need to multiply that value by 6 to correspond to a circle of radius 6 :)

Answer 29

Oh yuck! Ok let's do it:)

Answer 30

It's not too bad.
You have a fraction being divided by 6,
and you're multiplying that by 6.
So we just lose our denominator, ya?

Answer 31

yup!

Answer 32

So for part 1, I suppose we end up with 7pi.
Ok not too bad :o

Answer 33

Oh maybe I confused parts 1 and 2.. hmm thinking...
because now they're asking about the `tip` of the minute hand... mmm

Answer 34

Ooo boy, this is a poorly worded question :(
But `I think` what they want is 7pi/6 for Part 1,
and 7pi for part 2, we multiply the radian measure by the radius to get the arc length.

Answer 35

Mmm what do you think?
You as confused as I am? 0_o

Answer 36

Answer is 10pi/3:)