Question

Help. Showing working out thanks.

1) Express the relation x2 = y2 = 36 as two functions and state the largest domain and range of each.

2) 10 Circular ripples are formed when a water drop hits the surface of a pond. If one ripple is represented by the equation x2 = y2 = 4 and then 3 seconds later by x2 = y2 = 190, where the length of measurements are in centimetres,

a find the radius (in cm) of the ripple in each case

b calculate how fast the ripple is moving outwards.

(State your answers to 1 decimal place.)

3) Express the relation x2 + (y − 2)^2 = 9 as two functions stating the largest domain and range of each.

Answer 1

is it possible that in 1 and 2 it's x^2+y^2 and not x^2=y^2?

Answer 2

That's what I am thinking. He hasn't been terribly clear. It could also be x^2-y^2 which would make more sense as that is a circle.

Answer 3

the only reason why iam am asking is because 2 and 3 definitely involve circles

Answer 4

but he left so why waste our time?

Answer 5

\[x^2+y^2=36\] a circle with center (0,0) radius 6
solve for y get
\[y^2=36-x^2\]
\[y=\pm\sqrt{36-x^2}\] two functions are
\[f_1(x)=\sqrt{36-x^2}\] and
\[f_2(x)=-\sqrt{36-x^2}\]

Answer 6

Express the relation \(x^2 + (y − 2)^2 = 9\) as two functions stating the largest domain and range of each.
again a circle, this one with center (0,2) and radius 3
again solve for y
\[(y-2)^2=9-x^2\]
\[y-2=\pm\sqrt{9-x^2}\]
\[y=\pm\sqrt{9-x^2}+2\] two functions are
\[f_1(x)=\sqrt{9-x^2}+2\] and
\[f_2(x)=-\sqrt{9-x^2}+2\]

Answer 7

ok thank you so much but what can u also state largest domain and range? and the only reason I left was because I had to go to sleep and I wanted to know the answer the following day, which is today, I apologize for leaving.

Answer 8

I also believe that x^2+y^2 is the same as x^2=y^2 and the question from my textbook does clearly say x^2=y^2 so yeh i didn't type it incorrectly.

Answer 9

was this the problem? and what did you need help with exactly (it looks like there are already answers here)?

Answer 10

um i need help with the domain and range for question 1&3 and question 2 isn't solved.

Answer 11

is #1 x^2+y^2 = 36?

Answer 12

If so, then the domain is [-6,6]. We know this because the center x value is x = 0. We can then go 6 units in either direction along the x axis.

Answer 13

yep, i believe so. I was also confused by that but that is how it is written in my text book.

Answer 14

The range is the exact same way (just along the y axis now) and it is [-6,6]

Answer 15

hmm how odd...

Answer 16

likely a typo since I see this kind of typo all the time (and because the = sign is on the same key as the + sign)

Answer 17

not sure but question 2 also has it.

Answer 18

# 3

Domain: [-3,3]

This is because the center x value is 0 and we can go 3 units in either direction along the x axis

Range: [-1,5]

The center y value is y = 2. The radius is 3. So we can go as low as y = 2-3 = -1 and as high as y = 2+3 = 5

Answer 19

and this a question out of context but how do u keep posting comments without leaving this post? each time I reply I have to leave again and come back then i can post another comment.

Answer 20

hmm very strange, well the fact that they're talking about circles and their equations leads me to believe that the = signs have to be + signs

Answer 21

I'm not sure, it just works for me like it should. Perhaps there's a glitch in your browser?

Answer 22

i don't think so? It's always been like this and i'm not sure what to do? since it's pretty annoying. I'm using firefox atm.

Answer 23

try using google chrome, firefox gets buggy imo sometimes

Answer 24

you might have an extension or an addon running that doesn't agree or work with some feature on this site

Answer 25

yep ok.

Answer 26

google chrome is working fine.

Answer 27

ok great

Answer 28

thanks! :)

Answer 29

Sure thing, sounds like my suspicions were somewhat true. Your best bet to fix the issue is to disable/enable addons until you can identify the culprit.

Answer 30

yeh but I don't how to find which addon is the problematic one so yeh I guess i would just use google chrome for openstudy.

Answer 31

alright that works too

Answer 32

did I answer your math question?

Answer 33

um yep but could u answer question 2 and u got one of the range incorrect for #3. I have the answers, I will post them.

Answer 34

oh i see, they're breaking things up into two functions, one second

Answer 35

the answers also stated them separately, yep.

Answer 36

because it said state the largest domain and range for each.

Answer 37

for #3, solve for y


x^2 + (y-2)^2 = 9


(y-2)^2 = 9 - x^2


y-2 = sqrt(9 - x^2) or y-2 = -sqrt(9 - x^2)


y = 2 + sqrt(9 - x^2) or y = 2 - sqrt(9 - x^2)


So after solving x^2 + (y-2)^2 = 9 for y, we get y = 2 + sqrt(9 - x^2) or y = 2 - sqrt(9 - x^2)


Because the piece from y = 2 to y = 5 corresponds to the upper half of the circle, this means that the range of the upper function is [2,5]


Similarly, the piece from y = -1 to y = 2 on the original circle corresponds to the lower function's range, so the range of the lower function is [-1,2]


Hopefully this clears things up.

Answer 38

yep it does, thanks so much but can u help me with question 2?

Answer 39

the radius of the smaller circle is 2 cm (since sqrt(4) = 2)

the radius of the larger circle is sqrt(190) = 13.784 cm

So the radius changes from 2 cm to 13.784 cm in 3 seconds

which means that the speed is...

speed = (change in position)/(change in time)

speed = (13.784-2)/(3 - 0)

speed = (11.784)/(3)

speed = 3.928

Therefore, the speed of the ripple is 3.928 cm/s, you can round this to 3.9 cm/s as shown in the answer sheet.

Answer 40

thanks so much! I wish I was as smart as u ==''

Answer 41

You're welcome. Just keep practicing and you will become as smart as me someday, trust me (I was in the same position as you once).

Answer 42

haha I will keep practising and I don't think I will ever become as smart as you. ur Jim btw right?

Answer 43

Yes, I'm Jim. Nice to meet you Jay. Also, I'm sure you will (you just have to believe in yourself).

Answer 44

yep nice to meet u too Jim. I do try to believe in myself but I just never get it for some reason lol.

Answer 45

u should become a tutor btw.

Answer 46

well just keep at it and it'll click eventually, and funny that you mention it, I am a tutor lol

Answer 47

haha lol. that's great!

Answer 48

The best thing to do is to translate everything into terms you can understand. Once you've done that, things should be easier.

Answer 49

lol yes it is, I love it

Answer 50

yep I try to but I just somehow can't interpret it correctly? or sometimes I get part of it and don't get the other half of it. It's great that u love ur job. These are the type of questions that I get stuck on, like I do manage to do some of them but yeh.

Answer 51

which one do you want to do first?

Answer 52

wait um I was doing them in class b4 so I need to check my exercise book to see which ones I was stuck on and couldn't do. Wait a sec.

Answer 53

alright

Answer 54

Well question 5 i don't understand the "$x, where 21 601 ≤ x ≤ 58 000" part and I was having some problem with the Domain and Range of question 6 and 7.

Answer 55

let's start with #5 then

Answer 56

yep k

Answer 57

# 5

a)


They let x = taxable income (this the amount of money that the gov is allowed to tax)


and they let T = tax payable (this is the amount of money owed to the gov)


Since the tax for someone earning between $21600 and $58000 is $6264 plus 30% of anything over $21600, this means that no matter what, the person earning this much money has to pay at least $6264 to the government


So right now, we know that T = 6264 regardless of the value of x


Now add on 30% of the dollar amount over $21600, this means that if the person is making $21605, then the government is adding 30% of $21605 - $21601 = 0.3*$4 = $1.2 to the total tax bill


In general, if the person is making x dollars, and x is between 21601 and 58000, then the government is adding on 0.3(x - 21601) to the final tax bill


So sum all this up to get


T = 0.3(x - 21601) + 6264


now simplify


T = 0.3x - 0.3*(21601) + 6264


T = 0.3x - 6480.3 + 6264


T = 0.3x - 12744.3


So if someone is making $x and x is between $21601 and $58000, then the amount that s/he has to pay in taxes is T = 0.3x - 12744.3

Answer 58

oh i made a typo, one sec

Answer 59

wait for ur last step T = 0.3x - 12744.3 it's meant to be T = 0.3x -216

Answer 60

yes you caught it before I did lol

Answer 61

lol

Answer 62

should be T = 0.3x - 216.3

Answer 63

lost a sign back there and added them by mistake

Answer 64

yup k, no worries.

Answer 65

# 5

b)

Plug in x = 0 to get T = -216.3, so we have one point (0, -216.3)

Plug in x = 1 to get T = 0.3(1) - 216.3 = -216 giving us another point (1, -216)


Now plot the two points and draw a straight line through them to graph T = 0.3x - 216.3

Answer 66

To be more realistic, you can plug in values of x that lie in the given interval [21601, 58000] to get your two points

Answer 67

yep but they actually only used two x points 21601 and 58000

Answer 68

yeh

Answer 69

alright, you can certainly use those points as well

Answer 70

any two work

Answer 71

# 5

c)

Plug in x = 32000 and evaluate


T = 0.3x - 216.3

T = 0.3(32000) - 216.3

T = 9600 - 216.3

T = 9383.70


So the tax paid is $9383.70

Answer 72

x+4≤10
x≤6

a) P = x + 4 + x + 4 + x - 1 + x-1
= 4x + 6
b) I thought the domain would be (0,6]
and the range (6,30] but it's not.

Answer 73

# 6

a)

Perimeter

P = 2L + 2W

P = 2(x+4)+2(x-1)

P = 2x+8+2x-2

P = 4x+6


So the perimeter function is P = 4x+6


The values of x must be at least one because if x = 1, then x-1 = 0. Anything smaller than that for x will make a negative width, which doesn't make sense.


So we know that x > -1 (we're excluding -1 to make sure that we don't get a 0 width either)


but...we know that the max side length is 10 m, so we know that x+4 can't exceed 10 meters. This means that x + 4 <= 10 or x <= 6


Combine the two to get -1 < x <= 6


So the domain in interval notation is (-1, 6]


The range is straightforward once you know the domain. The smallest value in the domain corresponds to the smallest value in the range (similar with the largest values)


So the smallest range value is P = 4(-1)+6 = 2 and the largest range value is P = 4(6)+6 = 30


So the range is (2, 30]

Answer 74

oh wait, made another typo....doh

Answer 75

domain should be (1, 6] which makes the range to be (10,30]

Answer 76

for some reason, a negative popped up in my work somehow

Answer 77

well unfortunately I'm getting tired (really early in the morning over here lol), so I'm signing off, hope this helps

Answer 78

haha k, go rest or something! thanks for helping me. tty next time Jim, it was really nice meeting u.