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?
show all of your work

Answer 1

Let's paraphrase this Part 1: a) What is the "central angle" when the minute had traces one complete circle? b) What is the central angle when the tip of the minute hand moves from time 1:20 to time 1:55? c) Use the 'arc length' formula, s = r * theta, to answer the question posed in Part !.

Answer 2

ok. now how do i find the number of degrees per minute

Answer 3

Part 1: How many radians does the minute hand move from 1:20 to 1:55?

can you answer how many minutes (of time) the hand moved between 1:20 and 1:55 ?

Answer 4

35 minutes

Answer 5

a full circle is 60 minutes. what fraction is 35 minutes of the full circle ?

Answer 6

hmmm i know its over half

Answer 7

fraction will be part over whole

Answer 8

so probably 5/8? or is it a larger fraction

Answer 9

35 minutes out of 60 minutes, written as a fraction

Answer 10

i think its 7/12

Answer 11

it's the same idea as if you had 3 spoons, and you lose 2 of them. you lost ⅔ of your spoons. part/whole or 2/ 3

here 35/60

Answer 12

yes, 7/12

next idea: there are 2 pi radians in a full circle. you have 7/12 of a full circle
how many radians do you have?

Answer 13

or, if you like to first use degrees. There are 360 degrees in a full circle
and you have 7/12 of a full circle. how many degrees?

(then to get radians multiply by 2 pi/180)

Answer 14

***oops: (then to get radians multiply by pi/180)

Answer 15

210 degrees

Answer 16

if the fraction were ½, it's more obvious.
full circle is 2 pi radians
½ of that is ½ * 2pi = pi

if you have ¾ of a circle, you have ¾ * 2 pi = 3 pi/2

yes 210º

Answer 17

and then i multiply that by pi/180?

Answer 18

yes

Answer 19

you should get 7/12 * 2 pi = 7pi/6 radians

Answer 20

ok... i got 7pi / 6

Answer 21

ok good

Answer 22

btw, their hint meant:
1 minute is 1/60 of a full circle. in degrees 1/60 of 360º= 6º per minute
then multiply 35 minutes times 6º/minute to get 210º

that works too (obviously)

Answer 23

ooh ok i didnt see that lol

Answer 24

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

the minute hand's tip moved along the circumference of the circle.
It went 7/12 of "all the way around"
so one way to do the problem is figure out the circumference (using the very famous C= 2 pi r formula), then multiply by 7/12

or , because we know the angle in radians, we use the simple formula:
arc= r theta, where theta is the angle in radians

Answer 25

ok lemme finish filling out part 1 real quick

Answer 26

ok done now lemme look at pt. 2

Answer 27

what's a lemme?

Answer 28

"let me" but its shorthand

Answer 29

oh

Answer 30

ok in the C=2pi r... our r is our radius correct

Answer 31

yes

Answer 32

would our radius be 6 inces because it says she notices the minute hand is 6 inches long

Answer 33

the radius of a circle is the distance from the center to the circumference.

the minute hand starts at the center and goes to the circumference.
its length is the radius

Answer 34

ok thats what i thought. so its
C= 2(pi)(6)

Answer 35

yes, which simplifies to 12 pi

Answer 36

then i multiply that by 7/12

Answer 37

yes, because that is the fraction of the whole circumference the hand moved

Answer 38

i got 7pi

Answer 39

yes.

based on the question, they may want you to use the formula
\[ s = r \ \theta \]
where r is 6, and theta is the angle in radians (from Part I)
s is the length of the arc in inches.
try that way.

Answer 40

our angle for theta would be 35?

Answer 41

no, theta is the answer to part I.

Answer 42

ok so our 7pi/6? or the 210

Answer 43

remember, the minute hand moved from 1:20 to 1:55 which is 35 minutes
which is 35/60 of 360 degrees , which is 210º, which is 7pi/6 radians

Answer 44

the formula only works for theta in *radians*

Answer 45

oohhhh ok gotcha

Answer 46

ummm i got 7pi? is that right

Answer 47

does that match what you got doing it the other way ?

Answer 48

what do you mean by the other way?

Answer 49

scroll up

Answer 50

yeah i did it in s=r(theta)

Answer 51

you also did
C= 2 pi r = 2 * pi*6= 12 pi
then 7/12 * 12 pi = 7 pi

Answer 52

yes and i got the same answer for both

Answer 53

The first way is more common sense. the second way is faster if
(1) you remember the formula
(2) have theta in radians

Answer 54

ok id have to say it did seem quicker. alot less to write and was simpler

Answer 55

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

Answer 56

almost done filling out pt. 2

Answer 57

done

Answer 58

For part 3, I would use
s=r θ

they give you s (the distance the minute hand moved), and r= 6
so you can find theta (it will be in radians)

Answer 59

would our s be 35?

Answer 60

35 is minutes.
we are now looking at a circle with a radius of 6 inches
and talking about moving so many inches around it (i.e. along its circumference)
they told us how far we moved.

Answer 61

which was 7/12 or 210 degrees?

Answer 62

yes, but part 3 has moved to a different question.
Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 5π inches?


in other words, the minute hand moved 5π inches

Answer 63

soo multiply 5pi to see how many inche it moved?

Answer 64

the key thing is it moved *inches*. we could get a ruler and try to measure along the circle about 15.7 inches, to see where it ended up

but they use the goofy 5 pi (instead of a number) so the numbers will work out nicer.
So keep the distance as 5 pi (rather than 15.70796327....)

Answer 65

The idea is the *length of the arc* is measured in inches
we use
s=r θ

Answer 66

s=r θ

they give you s (the distance the minute hand moved), and r= 6
so you can find theta (it will be in radians)

Answer 67

ok i had to step away for a sec lemme catch up real quick

Answer 68

ok so s would be 5 pi = 6(7pi/6)

Answer 69

almost. they gave you s and r, but we don't know theta for the s they gave us. Remember: this is a different problem

Answer 70

yep thats right i keep forgetting its a different problem so how would we find our theta

Answer 71

first write down the correct equation

Answer 72

ok

5pi = 6(theta)

Answer 73

any ideas how to "solve" for theta?

Answer 74

use these two ideas:
1) 6/6 is 1
2) if we divide one side of the equation by 6, we also divide the other side by 6

Answer 75

i cant find anything about it in my notes but i am missing 2 pages... i need to keep better notes lol

Answer 76

ok i get that

Answer 77

so what we do to one side we do to the other

Answer 78

yes. and in this case we want to divide the right side by 6 because it makes the 6/6 = 1

Answer 79

so the didvide 5(pi) by 6?

Answer 80

yes. not hard write 5pi/6
simple!

you get
5 pi/6 radians = theta
(you should put in the units so we don't get confused about inches, degrees or radians)

remember this formula
s= r theta assumes theta is in *radians*

Answer 81

so 5pi divided by 6 and ur answer is theta or theta is 5pi/6 the equation

Answer 82

can you be more clear?

Answer 83

i know it confused me

Answer 84

From the beginning (of Part 3)

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

that says the minute hand moved 5 pi inches (along the arc)
they want to know what angle that is (but in radians not degrees)

we use
s = r theta where s is the arc in inches, r is the radius in inches, theta is the angle in radians

Answer 85

ok theta be the sum of the nequation 5pi/6 or would theta be the equation 5pi over 6

Answer 86

you found

5 pi/6 radians = theta

in other words, the angle theta is 5 pi/ 6 radians
that is the answer.

Answer 87

How many radians on the unit circle would the minute hand travel? 5 pi/ 6 radians

Answer 88

ok so it would be 5pi = 6 (5pi/6)

Answer 89

that is a true equation (simplify the right side to get 5 pi)

but the idea is
1) we want to find the angle theta
2) we found
theta= 5 pi / 6 radians

we found what the angle is. use its value as the answer to the question
How many radians on the unit circle would the minute hand travel?

Answer 90

ok so part 3 answer is 5pi/6 radians

Answer 91

yes

Answer 92

ok sorry im at work right now while doing classes lol. but im back

Answer 93

@mathmale

Answer 94

one last part