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?
Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 5π inches?
Part 4: What is the coordinate point associated with this radian measure?

Answer 1

I have to show all my work as well

Answer 2

@phi

Answer 3

I'll fan and give a medal

Answer 4

@ganeshie8

Answer 5

How many minutes from 1:20 to 1:55 ?

Answer 6

35

Answer 7

you the minute hand takes 60 minutes to go all the way around the circle.
what fraction of the whole circle is 35 minutes?

Answer 8

7/12

Answer 9

yes 35/60 which simplifies to 7/12 (divide top and bottom by 5)

any idea how many radians in a full circle?

Answer 10

2pi

Answer 11

I think

Answer 12

yes 2pi (roughly 6.28 radians in a circle)

now we think: the minute hand moved 7/12 of the way around the circle, so it moved 7/12 of 2 pi radians

Answer 13

1 and 1/6pi

Answer 14

yes, but people probably keep it an improper fraction
7/12 * 2 pi= 14 pi/12 = 7 pi/ 6 radians

Answer 15

btw, if we wanted degrees it would be 7/12 * 360 = 210º
(and we could change degrees to radians by multiplying by pi/180 ) we would still get 7 pi / 6 radians

Answer 16

ok on Part 1?

Answer 17

Part 2: How far does the tip of the minute hand travel during that time?

you know its 7/12 of the whole way round.
find the distance around the circle. i.e find the circumference C= 2 pi r
then find 7/12 of C

Answer 18

Oops! Sorry about the wait. How would we go about finding the radius for that?

Answer 19

you read the question carefully?

She notices that the 6-inch-long minute hand

Answer 20

*facepalm* I'm sorry. I missed that. In that case, 12pi (for the Circumference), and the distance would be 84pi/12, which simplifies to 7pi. Right?

Answer 21

But wait.. That would theoretically be over 3 hours, right?

Answer 22

That doesn't make sense.

Answer 23

remember that when we find the circumference we are finding the number of inches around the clock, so 7 pi inches (not radians, nor degrees)
roughly 22 inches out the all the way around (37.7 inches)

btw, I would have done it
\[ \frac{7}{12} \cdot 2 \cdot 6 \cdot \pi = \frac{7}{\cancel{12}} \cdot \cancel{12} \cdot \pi = 7 \pi \text{ inches}\]

in other words, rather than multiply everything out and then dividing by 12, divide by the "little" numbers... it is easier

Answer 24

For Part 2, we could use the formula
\[ s = r \ \theta\]
where s is the length of the arc, r is the radius, and theta is the angle *in radians*
if you remember that, and use Part 1 (i.e. theta= 7 pi/6 ) you would get
s = 6 * 7 pi/6 = 7 pi (same answer!)

Answer 25

Alright! Thank you! So- Part 3

Answer 26

Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 5π inches?

I would use the convenient formula
s = r * theta

can you do that ?

Answer 27

s=6*7pi/6=42pi/6=7pi

Answer 28

that is for Part 2.

what about Part 3 ?

Answer 29

and btw, 6*7pi/6 is the same as (6/6) * 7 pi = 1* 7pi = 7pi
no need to multiply out 6*7 to get 42. (notice you can do 6/6 in your head)

Answer 30

s=6*5pi

Answer 31

Right? I'm sorry- I'm confused

Answer 32

Which is 30pi

Answer 33

You have to read the question and "match up" the words to the symbols in
s= r theta

it helps to remember that s is the length of the arc (along the circumference)
r is the radius, theta is the angle in radians

Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 5π inches?

They are asking about *radians*. that means *angle* i.e. they want you to find theta
they say the minute hand moves 5 pi inches. Inches is a clue we are moving along an arc.
in other words, they are telling you what s is
and of course, the radius r is 6 inches

Answer 34

So i need to rearrange the equation to be theta=s/r

Answer 35

Which is theta= 5pi/6

Answer 36

yes, that is one way to proceed. Or you replace the variables with the numbers that you know, and then solve for theta.

yes. theta= 5pi/6 radians (you should get in the habit of labeling the units.)

Answer 37

Ok- thank you. We're so close! :)

Answer 38

Part 4: What is the coordinate point associated with this radian measure?

can you plot angle 5pi/6 on the unit circle?
(I find using degrees is easier)

Answer 39

Couldn't I just pull up a picture of the unit circle and check 5pi/6? I know that's a bit of a shortcut, but that's what the course has shown me thus far

Answer 40

yes, you could. But it is not hard to figure out. 5 pi/6 * 180/pi = 150º