Cool
#help-13
blissful turtle
and the 1/pi is brought down
woeful isle
All that to get 1
woeful isle
and the 1/pi is brought down
Ye
blissful turtle
looks it
i reckon its gonna give pi
thanks so much for being so patient and helpful
woeful isle
Algebraic expression
blissful turtle
this was a lot of fun
woeful isle
Doubt it will yield pi
woeful isle
thanks so much for being so patient and helpful
Appreciate it
woeful isle
this was a lot of fun
Truly was
cedar kilnBOT
cedar kilnBOT
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cedar kilnBOT
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weak jungle
im kinda confused with negative angle identities
cedar kilnBOT
im kinda confused with negative angle identities
golden owl
what is with the odd and even
Give an example
dusk goblet
tan(-x) = -tanx
weak jungle
yeah btu how do u apple that here
dusk goblet
its 1.5 rright?
yes
weak jungle
apply
dusk goblet
$\tan(-x) = \frac{\sin(-x)}{\cos(-x)} = -\frac{\sin x}{\cos x}$
wraith daggerBOT
knief
dusk goblet
since sin is odd and cos is even, tan is odd
weak jungle
can u give me an example
dusk goblet
🤔
i’m not sure what you’re confused about
do you know the definition of odd/even functions
weak jungle
no
weak jungle
my teacher never explained what thjey were they just said this is odd and this is even
oh ok
.close
dusk goblet
an even function satisfies the following : f(-x) = f(x)
cedar kilnBOT
cedar kilnBOT
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dusk goblet
🤔
dusk goblet
an even function satisfies the following : f(-x) = f(x)
geometrically this means that the function f is symmetric about the y axis
an odd function satisfies f(-x) = -f(x)
which geometrically means is symmetric about the origin
looking at the graph of a few odd/even functions might help your intuition
weak jungle
so if its cos(-2) its cos(2) and what if it is cos(2)? is it cos(-2)
dusk goblet
what
dusk goblet
who said anything about changing numbers
dusk goblet
no
weak jungle
huh
dusk goblet
2 did not go to -2
you simply identified that when 2 is the input of cos it yields the same result as when -2 is
dusk goblet
yes f(-x) = f(x)
it is symmetric about the y axis
notice that -2 is the reflection of 2 in some sense about x = 0
weak jungle
what
dusk goblet
,w plot x^2
wraith daggerBOT
dusk goblet
if we folded one side over the y axis (x=0) they would line up perfectly
weak jungle
hm okay
and for odd functions, they are symmetrical about the origin
what wpudl tahgt look like
would that*
dusk goblet
,w plot tanx
wraith daggerBOT
weak jungle
hm so like its flipped over?
dusk goblet
yes
weak jungle
isnt that for cos too
dusk goblet
,w plot x^3
weak jungle
,w plot sinx
wraith daggerBOT
wraith daggerBOT
weak jungle
doesnt this line up too?
dusk goblet
nope
if we flipped the left portion over the x axis then flipped that again over the y axis then it would line up
dusk goblet
for instance, if we just flipped the left portion of sinx over the y axis it would look like -sinx for x > 0 (to the right of the y axis)
,w plot -sinx
wraith daggerBOT
dusk goblet
ignore the left part here
dusk goblet
the right side is what we’d get after folding the left side from sinx over the y axis
dusk goblet
but notice this is still flipped so introducing the - in front will make it line up with sinx
hence
-sin(-x) = sinx
🤯🤯
,w plot -sin(-x)
wraith daggerBOT
weak jungle
i understand it now
dusk goblet
visuals usually help
dusk goblet
you’re welcome
weak jungle
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weak jungle
oh my god im so sorry but i understand even but odd is still confusing. i know the answer of odd functions i just dont see how you get there
weak jungle
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cedar kilnBOT
weak jungle
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frigid dust
whats the question?
hollow trail
an odd function is a function f such that f(-x) = -f(x)
weak jungle
yeah i get that part but
i dont understand how it id
i know it is but i dont how how it is
if that makes sense
mental trail
like what does it represent?
mental trail
if you plot an odd function
weak jungle
you flip it along the origin
mental trail
and decide to put a pin on the origin
and rotate it 180 degrees
the function will remain unchanged
weak jungle
yeah
twilit escarp
What is your trouble with odd function then ?
weak jungle
why will it remain unchanged
mental trail
? because we said so
weak jungle
If cosec x = -5, find cosec (-x)
mental trail
we said "odd functions are the ones that behave like this"
frigid dust
if you flip an even function on the y axis it is also unchanged
mental trail
that's how we defined "odd functions"
mathematically, we define odd functions to be: f(-x) = -f(x) for all x
mental trail
and decide to put a pin on the origin
graphically, it's this
weak jungle
hjm
mental trail
so
if you want to find cosec(-x) knowing cosec(x) is -5
well
KNOWING that cosec = 1/sin is odd
hollow trail
if you flip it across both the x- and y- axes then that's the same as rotating it 180 degrees about the origin. for a function f(x), -f(-x) is that function flipped across the origin in that manner. and the definition of an odd function says that -f(-x) = -(-f(x)) = f(x), i.e. it is unchanged from that transformation
mental trail
cosec(-x) = -cosec(x)
hollow trail
another way to think about it is that flipping an odd function across the x-axis and flipping it across the y-axis has the same result
weak jungle
i understand it grraphically just not mathematically
frigid dust
sometimes its useful to have a property like an odd function
same with odd and even numbers
weak jungle
for even i understand it mathematucally but not graphically
frigid dust
knowing a number is even, multiplying it with an even number you get an even number
which can sometimes be useful property
you have the same with functions
*similar
mental trail
knowing a number is even, multiplying it with an even number you get an even num...
multiplying with any integer as well
frigid dust
so if you have a function, and notice that its odd or even, you can apply properties of odd and even functions which can sometimes be useful
if that makes sense
weak jungle
If cosec x = -5, find cosec (-x) can someone explain this example
weak jungle
so if you have a function, and notice that its odd or even, you can apply proper...
oh ya
that makes sense
but my dumb brain cant do that
wait
yes it can
oh
ok that helped but if someone explains If cosec x = -5, find cosec (-x) i think that would be better
hollow trail
If cosec x = -5, find cosec (-x) can someone explain this example
if you know whether cosec is odd or even then you can find cosec(-x) based on the known value of cosec(x)
weak jungle
yeah its odd
frigid dust
dones thave to be just cosec
hollow trail
so use the definition of an odd function
frigid dust
given f(3)=-5 and the function is odd, what is f(-3)?
weak jungle
5?
frigid dust
yup
weak jungle
hm okay\
thank you guys for being patient
and helping
im not gonna close just yet in case if i have trouble with one more but i should be fine
frigid dust
is sin*cos even or odd?
weak jungle
ye
mental trail
yuyp
:(
unfortunately the analogy between even/odd functions and even/odd integers doesn't go far
so don't rely too much on it
hollow trail
in functions we have:
even*even=even
odd*odd=even
even*odd=odd
even + even = even
odd + odd = odd
even + odd = neither
weak jungle
yeah ive realized
frigid dust
ye, mb
weak jungle
?
frigid dust
whenever i use the properties i always make a little sketch, to make sure i get it right
sin*cos is indeed odd
weak jungle
wait wha
frigid dust
anyway, you get the point
weak jungle
how is sin*cos odd?
frigid dust
beacuse odd* even function is odd
weak jungle
how?
hollow trail
if f(x) is odd and g(x) is even then
(f*g)(-x) = f(-x)*g(-x) = -f(x) * g(x) = -(f*g)(x)
twilit escarp
consider f(-x) and -f(x)
wraith daggerBOT
weak jungle
what does -sin(-x) = to
digital cliff
sin(x)
weak jungle
omg
that makes so much sens
e
bro i should just graph it
ahahahah yes
im so happy thank you gusy for ur patience and all the help
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cedar kilnBOT
weak jungle
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hollow trail
you are not graphing one of them correctly then
weak jungle
oh
twilit escarp
,w graph cos(x)
wraith daggerBOT
twilit escarp
,w graph cos(-x)
wraith daggerBOT
weak jungle
,w plot -cosx
wraith daggerBOT
weak jungle
,w plot sin(-x)
wraith daggerBOT
weak jungle
it cant be
thats
impossible
when x is negative in sin then the orgin is going down hence when the x is negative in cos shouldnt it go up rather than positive x which goes down?
hollow trail
if you have a function f(x), then taking f(-x) flips it horizonally across the y-axis
hollow trail
it is flipped horizontally, and it ends up with the same plot because cos is even
,w plot e^x and e^(-x)
wraith daggerBOT
weak jungle
but thats just like circular reasoning
hollow trail
for an example of what a different function looks like flipped horizontally
you can know that cos is even without looking at the graph
weak jungle
yeah but i dont understand unless the graph shows me that
and i dont udnerstand how the grph got there
because if cos(-x) is cosx flipped horizontally shouldnt it start at the bottom?
weak jungle
,w plot sin(-x)
wraith daggerBOT
digital cliff
huh
weak jungle
so ur saying when it flips along the x axis that vertically?
digital cliff
'flipped horizontally' means reflected in the y axis
vertically is reflected in the x axis
weak jungle
but x is horizontal and y is vertical who came up with these words 😭
hollow trail
the axis is perpendicular to the direction of reflection
weak jungle
so when i graph cos(-x) what happens?
hollow trail
this is what you get if you plot f(x) and f(-x) for a function which isn't odd or even
weak jungle
ok but cos is even
hollow trail
there isn't any difference in how it's done, but you can't show both on the same graph because they are identical
weak jungle
,w plot tan(-x)
wraith daggerBOT
weak jungle
there isn't any difference in how it's done, but you can't show both on the same...
yeah but cos x flipped horizontally is not the same thing
weak jungle
how is that horizontally its up and down
hollow trail
this is an image of the plot of cos(x) flipped horizontally
hollow trail
how is that horizontally its up and down
the arrows are horizontal
weak jungle
this is an image of the plot of cos(x) flipped horizontally
i am so confused as to hjow thats horizontally and not vertically
hollow trail
because the pixels of the image were moved horizontally relative to their original location
hollow trail
the axis we mirrored about is vertical, but we call it a horizontal mirroring because the movement is horizontal
weak jungle
can you explain that logic in sin then
sorry
im just confused
like my whole life has been a lie
hollow trail
,w plot sin(x)
wraith daggerBOT
hollow trail
horizontal flip across the y-axis, what you would get if you plot sin(-x):
vertical flip across the x-axis, what you would get if you plot -sin(x):
weak jungle
vertical flip across the x-axis, what you would get if you plot -sin(x):
i understand how this is vertical
weak jungle
horizontal flip across the y-axis, what you would get if you plot sin(-x):
but now hoe this is horizontal
hollow trail
look at the writing, etc
weak jungle
wait im so dumb
OH MY GOD IM SO DUMB
OHHHH I UNDERSTAND
cloud
thank you so much
fir being patient and helpful
i appreciate you
🙂
i understand
im going to close the case now
oh it feels so good
bye bye cloud
.close
limber kiln
When it asks for the interval where f(x) is greater or equal to g(x), do I include the vertical asymptote of g(x)? In other words, would the values of x be between -13.675 and 3, or just greater than-13.675
trail portal
first option i guess
limber kiln
should i assume that for every greater-than question i get
idk im like really bad at remembering rules
celest flume
i would say first option as well
limber kiln
celest flume
you cant really compare functions when one is undefined
limber kiln
you cant really compare functions when one is undefined
yea thats kinda what i was thinkin
trail portal
always base on the domain
limber kiln
so would this be right
limber kiln
ah ok
celest flume
🦐?
limber kiln
and don't take 3 in the intervale
wait so it would just be x is greater than -13.674
trail portal
lemme find something to explain to you simple way
so you find the intervale of your fonctions
intersection*
or idk
the point where it's both same x and y
trail portal
stop a the second one
if there isn't a second one
it's stopping at the domain of the lower function
limber kiln
alright thanks
trail portal
alright thanks
so explain it by yourself
limber kiln
the function is greater after the intersection of the points
and the value of x that satisfies the expression is between the intersection and the point where either is undefined
right?
trail portal
i guess
find just two intersections
if there is only one
check the variation after this point
in our case g(x) was lower than f(x)
and stop at the bounds of the domain of the lower one
what I'm saying
hold on
just do A ∩ B
trail portal
but you need to know how
limber kiln
i just wanted clarification on whether or not the asymptote was included in the statement
trail portal
asymptote isn't the problem i think
you see the domain?
if there is no second intersection you'll stop at the end of domain of g(x) or f(x)
limber kiln
in this case ill stop at g(x)
trail portal
yeah
limber kiln
because any more than that, it wouldnt fit the domain of the overall expression
alright
limber kiln
f(x)
trail portal
yeah that's my boi
trail portal
no need apologies
limber kiln
.close
cedar kilnBOT
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icy spear
Can someone help do this question and show the solving out? The mark scheme makes no sense nor does the yt vids
cedar kilnBOT
Can someone help do this question and show the solving out? The mark scheme make...
@icy spear Has your question been resolved?
zenith sedge
how is there a graph of log(-x) if u cant take the log of a negative number
pastel vault
how is there a graph of log(-x) if u cant take the log of a negative number
exactly
so when x is already negative, -x will be positive
this implies the domain is -x > 0 or x < 0
cedar kilnBOT
pastel vault
no worries
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cloud creek
A half-full bottle of water has exactly as much as four identical empty bottles. We replace the bottle on the right tray with a full bottle of water.
How many empty bottles will he have to place on the left tray to balance the scales?
cloud creek
i am 90 percent sure the answer is 8
but i have a weird feeling that this is a trick question 😭
undone epoch
How did you get to that answer?
cloud creek
well i just multiplied the left quantity by two as the right quantity was multipled by 2
oh wait
is it 6
undone epoch
Multiply by two? What is the situation on the right, then?
cloud creek
wdym
undone epoch
On left, you get 8 empty bottles. On right, is it 2 half bottles or a full bottle?
cloud creek
oh
oh since i add 1/2 on right to get a full bottle
on the left
i should add 1/2 of 4
to get 6
therefore 1 full bottle is equivalent to 6 empty ones
undone epoch
You should know how many empty bottles half water (without bottle) is.
cloud creek
i don't undersntad
undone epoch
4 bottle = 1 bottle + 1/2 water
cloud creek
right
undone epoch
1 water = 6 bottles, you got it right here, but the water needs to be stored in a bottle.
cloud creek
so its 7?
oh
so ur saying 1/2 full water
inside a bottle is 4 as shown, so 1/2 full water by itself is equivalent to 3 empty bottles
oh
i knew i missed something
thank you
undone epoch
All good.
vague creek
A half-full bottle of water has exactly as much as four identical empty bottles....
if:
4 empty = 1 bottle plus half
3 empty = 1 half
6 empty = 1 whole
cloud creek
yes
cloud creek
if:
4 empty = 1 bottle plus half
3 empty = 1 half
6 empty = 1 whole
are you trying to add something or was this jsut to clarify things
cloud creek
so you do agree 7 is the right answer
vague creek
yes i do for now
cedar kilnBOT
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ornate sinew
how to justify that a figure created by revolving a triangle along one of its sides will have the largest volume when the triangle has the largest area
cedar kilnBOT
@ornate sinew Has your question been resolved?
cedar kilnBOT
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cunning prawn
This is more of an economics question but it does have some math and i dont know where to go for help
cunning prawn
I’m struggling on understanding what it is asking me to do
worldly chasm
Do you understand what the index column is showing you?
cunning prawn
not really im assuming its percentage change
worldly chasm
in other words, if you buy something in 2010, it costs 110.1/100 times more in 2013
if you buy something in 2012, it costs 104.5/107.4 times as much in 2011.
it's a way of relating the values in the other columns against each other in a way that compares apples to apples
cunning prawn
am i meant to compare 2010 and 2013 both ways for each collum
worldly chasm
the idea is you convert all of the numbers to 2010 dollars, and compare them. Then convert all of the numbers to 2013 dollars and compare them.
inner solar
cedar kilnBOT
inner solar
why is this function continous on set [a,b] and why does it have derivative on (a,b)?
crimson delta
the second part is obviously both of these
so really the question is whether f has those properties
crimson delta
the second part is obviously both of these
(as its linear)
inner solar
the second part is obviously both of these
thats what i dont see could you elaborate?
and yes f is both of those things because it is in the presupposition of the proof im doing
(its the proof of Lagrange's mean value theorem)
crimson delta
happens
cedar kilnBOT
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dawn mica
hey guys i need someone who can solve math problems in arabic (plz)
idle gulch
hey guys i need someone who can solve math problems in arabic (plz)
We all use arabic numerals right?
oblique flare
hey guys i need someone who can solve math problems in arabic (plz)
Send I might help
oblique flare
its about function ,and i am pretty sure u can guys solve it 🙂
Asks for the domain of the function
oblique flare
its about function ,and i am pretty sure u can guys solve it 🙂
مجال تعريف الدالة بمعنى القيم التي يمكن التعويض بها مكان x
idle gulch
We all use arabic numerals right?
What you call the original numerals are actually arabic !
dawn mica
i know its just some people think its from engish people
so , can anyone solve (ONLY ONE) and also teach me how to solve it so i could finish them all
oblique flare
so , can anyone solve (ONLY ONE) and also teach me how to solve it so i could fi...
The domain of a function is the set of values x that you can evaluate in the function
like for the function f(x)=1/x
we can’t evaluate it at x=0
Bcz it’s undefined
So the domain of the function f(x) is D=R-{0}
dawn mica
so u are telling me i should find what is x using random numbers
but its not the same as the equation
oblique flare
No not really
Take f(x)=1/x-2 for example
The denominator cannot be 0
so we find the value of x that make it 1/0 and we exclude it from the domain
oblique flare
i am sorry to take your time
like let’s say x=2 in the function
1/(2)-2=1/0 we can’t divide by 0
so at x=2 the function is undefined
The domain is the set of numbers of x value that is defined in the function
dawn mica
ok should i make the f(x)=0 or make it positive
supple pollen
ok should i make the f(x)=0 or make it positive
Not the case, typically you have to know at what x so that the function has a corresponding value (range), you do this by making a set notation
So for instance the domain of f(x)=sqrt(x) means the value of x so that f(x) has defined/corresponding value, this means x is equal or greater than zero. x can't be negative however
So the domain is {x | x>=0}
dawn mica
so its equation but in function right?
i really cant understand
maybe i will skip it
.close
cedar kilnBOT
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formal copper
Anyone please help me with partial frac
cedar kilnBOT
Anyone please help me with partial frac
@formal copper Has your question been resolved?
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cedar kayak
On a German highway, black cars appear four times more often than
cars of other colors. Most black cars are driven by men: 60% of men drive black cars, while 20% of women drive
cars of a different color. We observe the cars on the highway. Give the probabilities in percent, rounded to whole numbers.
What is the probability of observing a man driving a black car?
What is the probability that the car is not black?
We see a black car. What is the probability of a woman
driving it?
cedar kayak
I'm lost with this task
cedar kilnBOT
@cedar kayak Has your question been resolved?
golden owl
On a German highway, black cars appear four times more often than
cars of other ...
First, try to describe things with variables:
TC: Total of Cars
BC: Total of Black Cars
OC: Total of Other Cars
M: Total of Man
W: Total of Woman
MB: Total of Man that drive Black Cars
WB: Total of Woman that drive Black Cars
. . .
and that goes on
Then, describe the probability with those variables:
a) What is the probability of observing a man driving a black car?
Answer: We want the probability to the event E that is
E: Man driving a black Car
So, the probability will be
P(E) = Total of man that drive a black car / Total of man = MB / M
By the text, it follows that MB = 0.6M -> P(E) = MB / M = 0.6M/M = 0.6.
b) and c) goes on just like that, but you will have to use some other properties of probability
peak kettle
yo guys can someone help me understand "Proof by Induction: Sum of Squares"
dm me if you can help pls!!
golden owl
yo guys can someone help me understand "Proof by Induction: Sum of Squares"
dm m...
Try using another #help channel, this is already occupied by joris :v
cedar kayak
Is it possible to explain the given probabilitys with a tree diagram to visualize?
violet flume
Is it possible to explain the given probabilitys with a tree diagram to visualiz...
is there a reason you'd like to use a tree diagram?
id think that a categorical square might be more intuitive
cedar kayak
I never worked with a categorical square
violet flume
something like this @cedar kayak
maybe you start here, with the given information
cedar kayak
<@&286206848099549185>
cedar kayak
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cedar kilnBOT
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ripe yarrow
So dot product gives angle between two vectors
cedar kilnBOT
So dot product gives angle between two vectors
ripe yarrow
And cross peoduct outputs a vector perpedicular to the vectors being cross producted
So the dot product of the vector perpendicular and one of the vectors being dot producted would be pi/2?
hollow trail
the dot product would be 0
the dot product gives a way of calculating the angle between two vectors, it's not itself the angle between two vectors
jade charm
Dot product is just multiply componentwise and sum
ripe yarrow
Dot product is just multiply componentwise and sum
Intuitive-wise
hollow trail
the dot product has a formula given as [ \vb u \cdot \vb v = \norm{\vb u}\norm{\vb v}\cos\theta ] so the angle between two vectors $\vb u$ and $\vb v$ can be calculated as [ \theta = \cos^{-1} \ab(\frac{\vb u \cdot \vb v}{\norm{\vb u}\norm{\vb v}}) ]
ionic finch
So after doing the dot product
For intuition, how much is one vector pointing in the direction of other?
wraith daggerBOT
cloud
ripe yarrow
For intuition, how much is one vector pointing in the direction of other?
Ok i like this but
Wouldnt it be 0 if the vectors are opposing eachother?
hollow trail
it is 0 if the vectors are perpendicular (cos(90) = 0)
ripe yarrow
I guess it could still be zero if theyre perpendicular
hollow trail
it's positive if they point in roughly the same direction (the angle is acute) and negative if they point in roughly opposite directions (the angle is obtuse)
cedar kilnBOT
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leaden cloud
What does the book mean hear by “the locus of the corner is a straight line”?
I came to the conclusion that the answer was to draw across the sides of the triangles and then draw downwards on both sides to have two vertices touch the base and have two vertices touch each side
leaden cloud
I’m not confident that’s the right way to solve this problem though and I might be missing something
knotty shard
Hey!
fallen heath
Could you help me?
You know trigonometry?
fallen heath
What does the book mean hear by “the locus of the corner is a straight line”?
...
Use Fig 3, and mark all points with alphabets, ABC being the triangle, etc.
Show me what you got
leaden cloud
You know trigonometry?
I think I remember some of it
leaden cloud
Use Fig 3, and mark all points with alphabets, ABC being the triangle, etc.
Mark the square points too right?
leaden cloud
Alright
leaden cloud
What now?
Wait hold on I messed this up
Actually nvm I think it’s good enough to get my point across
@fallen heath I did it
fallen heath
Now, assume B is origin, A7A8 = x, BA7 = l. Notice that x = l tan B, and so, coordinates of point A6 is, (l + x, x) = (l + ltan B, ltan B)
Similarly, what is coordinates of A2? If BA3 = l'?
leaden cloud
Why do I need to do this to find the other two vertices that aren’t on the base of the triangle?
I don’t get it
cedar kilnBOT
@leaden cloud Has your question been resolved?
fallen heath
What does the book mean hear by “the locus of the corner is a straight line”?
...
I'm leading you to your Question
steel crest
What does the book mean hear by “the locus of the corner is a straight line”?
...
it means that if we ignore the condition of of that point being on a side, the set of possible corners, would be a triangle
fallen heath
"the locus of corner is a straight line", clearly if you write coordinates of A6 and A2, you'll notice that the lines BA6 and BA2 coincide, or in other terms, "locus of corner" will be a "straight line"
leaden cloud
Why am I able to make a square by doing that?
This just brought back the feeling of hopelessness that I always get with math, I know I shouldn’t be hard on myself because it was a type of math I haven’t done in years and it was more of an example for teaching something else
But still
I hate how stupid I feel
fallen heath
You draw that line, make it intersect AC at, say E. Drop perpendicular from E to BC at F. And mark G on BC such that EF = FG
And draw perpendicular to BC at G intersecting AB at D, to get your square DEFG
leaden cloud
DEFG?
fallen heath
Yes, the desired square inscribed in the triangle with all vertices on the edges of given triangle
fallen heath
So, that is why... understanding that the points B, A6 and A2 are collinear is crucial
As is said by the author
leaden cloud
Collinear?
I don’t even think I heard that
I think I’m gonna log off for the night
fallen heath
Let me rephrase that
fallen heath
So, that is why... understanding that the locus of the corner A6 of this square is a straight line is crucial..
That is, if you move A7 on BC such that A5 and A8 are on sides AB and BC respectively, the point A6 will follow a straight line
cedar kilnBOT
@leaden cloud Has your question been resolved?
lethal linden
can anyone help me solve this problem?
cedar kilnBOT
can anyone help me solve this problem?
carmine bronze
pi would cancel out, 2pi r = (a/b) pi r^2.
lethal linden
pi would cancel out, 2pi r = (a/b) pi r^2.
?
vestal bluff
wait i am stupid
lethal linden
a/ b = r/2 no?
vestal bluff
just find the radius of the circle
lethal linden
just find the radius of the circle
ik
but how lol
am i missing any observations here
dry dirge
connect O and C and with Pythagorean theorem you will find radius
lethal linden
i mean OC^2 = OB^2 + CB^2
but i only know CB here
dry dirge
CB=6?
vestal bluff
you can find OB in terms of OC
lethal linden
yeah
lethal linden
OB = (AB-AO)
OB = (10-r)
carmine bronze
And what is OC?
lethal linden
oh damn
missed it
this was easier than i expected
bruh
OC^2 = OB^2 + BC^2
r^2 = (10-r)^2 + 36
r^2 = 136 - 20r + r^2
r = 6.8?
lethal linden
but the fact is the answer should be an integer
carmine bronze
This is a step in the process. You need to find a/b first, then add a+b.
lethal linden
but isn't a = r?
carmine bronze
$2\pi\cdot r = \frac{a}{b} \pi r^2$
wraith daggerBOT
Kookiemon
carmine bronze
$2 = \frac{a}{b} r$
wraith daggerBOT
Kookiemon
carmine bronze
What do you need to do to solve for a/b?
lethal linden
just divide both sides by r
carmine bronze
Yes.
lethal linden
2/r = a/b
2 = a
b = r = 6.8
carmine bronze
Keep r in fractional form. It will make solving this easier.
lethal linden
it mentioned that a and b are co-prime tho
carmine bronze
You need to find the numerator, a, and the denominator, b, of 2/r.
What is 6.8 in fraction form?
lethal linden
oh got it
a = 5 and b = 17
17 + 5
when i was trying for the first time i did get r = 6.8 but forgot that i can just keep it as a fraction
bruh
carmine bronze
Realizing when you need to add on to what was given is a big part of solving these types of problems.
lethal linden
Realizing when you need to add on to what was given is a big part of solving the...
i did the adding part right, but did some silly mistakes when finding r
can i get help for another problem in this forum?
carmine bronze
Start a new channel and post your question there.
Don't forget to close this channel.
lethal linden
.close
vast ermine
Help please
cedar kilnBOT
Help please
idle tusk
Help please
so replace every $x$ in $\frac{2x}{x-6}$ by $10$ and simplify the result
wraith daggerBOT
artemetra
vast ermine
f(10) means "replace every instance of x you see with a 10"
Ohh okay
vast ermine
so replace every $x$ in $\frac{2x}{x-6}$ by $10$ and simplify the result
Thank you that's simple
How do I do b?
idle tusk
How do I do b?
you have that $y = \frac{2x}{x-6}$\
an easy way to invert the function is to swap x and y to get $x = \frac{2y}{y-6}$ and then solve for $y$ from this
wraith daggerBOT
artemetra
cedar kilnBOT
cedar kilnBOT
Available help channel!
tough schooner
Does anyone know the generalization of these types of problems? let n be the number of colors and k be the number of required pairs
tough schooner
also suppose there are an indefinite amount of socks of each color available
steel crest
assume you have 19 pairs, how many socks maximum can you take without getting 20 pairs
add 1 to that value
and you get the answer
tough schooner
so is the answer 39?
tough schooner
i did that manually. I got 44
tough schooner
is that the EXACT answer?
steel crest
yes
steel crest
2k
tough schooner
ohh yeah sorry
steel crest
yes
tough schooner
I was confused, thanks for clearing things up
tough schooner
so i suppose that the general form is gonna be (2k + n-1)
is this formula widely used tho? i couldn't find it anywhere
cedar kilnBOT
cedar kilnBOT
Available help channel!
eager steppe
cedar kilnBOT
eager steppe
I can translate if anyone needs me to.
pastel vault
what have you tried so far
for instance, $\sum_{k = 0}^{49} u_k = \frac{50}{2} (\text{first term} + \text{last term})$
eager steppe
Im fairly rusty when it comes to sums.
pastel vault
because you have $u_0$ instead of $u_1$
first term = $u_0$ and last term = $u_0 + 49r$
wraith daggerBOT
south
pastel vault
on the internet you'll see (n - 1)d cause they start from u1
(from 0 to 49 there are 50 numbers, the same as from 1 to 50)
eager steppe
I know u19 = u0 + 19r and that sum goes from n=0 to n=49
as such, if we subtract the 19 we know from the value of 19
pastel vault
yeah okay then I guess you're having trouble using algebra, what to do next
eager steppe
we get sum ranging from 20 to 49 = 25-19 = 6
wraith daggerBOT
south
pastel vault
**south**
so this gives us 25(u0 + u0 + 49r) = 25
2u0 + 49r = 1
but you already have u0 + 19r = 17
two variables, two linear equations
eager steppe
so this gives us 25(u0 + u0 + 49r) = 25
2u0 + 49r = 1
but you already have u0 +...
Ok just did the math this makes sense to me
eager steppe
so this gives us 25(u0 + u0 + 49r) = 25
2u0 + 49r = 1
but you already have u0 +...
Ok so my idea now is to seperate this equation to give us the other one
Ex:
2u0 + 49r = u0 + 30r + u0 + 19r
u0 + 30r + 17 = 1
one sec
pastel vault
yeah that also works
eager steppe
u0 + 30r + 17 = 1
We can go one level deeper
pastel vault
so you have u0 + 19r = 17 and u0 + 30r = -16
eager steppe
u0 + 19r + 11r = -16
pastel vault
interesting
pastel vault
yeah that is indeed correct
eager steppe
r = -3
then we can go back and use that to deduce the value of u0
Alright that solves it. tysm for your help
I ll review my sum formulas
pastel vault
,w sum (74 + (-3)r) from r = 0 to r = 49
wraith daggerBOT
pastel vault
,calc 74 + 19 * (-3)
wraith daggerBOT
Result:
17
No I meant I ll like revise them
pastel vault
I know
eager steppe
since I wasnt able to do this alone because I forgot about it.
pastel vault
yeah keep practicing, keep referring to your notes to help you
cedar kilnBOT
cedar kilnBOT
Available help channel!
past sleet
question 9 please
cedar kilnBOT
question 9 please
past sleet
here is my working out for all parts before part f and g
im not sure how to solve 9 g
the last bit of part e is incorrect i think i should ‘ve written v(t) = 0 after 18.5 seconds
since the displacement is 0 at 9.5 seconds
<@&286206848099549185>
e
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
sonic tusk
cedar kilnBOT
sonic tusk
so far i have that (a^2)c + ab^2 + bc^2 all over abc is equal to the integer n
im thinking of using contradiction?
but idk
first of all my combined fraction even looks weird so i don't even know if that is correct
assuming that its an integer
then i should be able to take a factor of abc out of the numerator
mighty shuttle
Maybe use the fact that the integers are closed under addition
mighty shuttle
or prove the contrapositive
pastel vault
that doesn't help, it has to relate to the fact you have a/b + b/c + c/a specifically
hmmm
mighty shuttle
Prove that a/b or b/c or c/a is not an integer, the a/b+b/c+c/a is not an integer
that's the contrapositive
that's easier
pastel vault
so far i have that (a^2)c + ab^2 + bc^2 all over abc is equal to the integer n
something like a^2 c + ab^2 c + bc^2 = nabc, hence LHS must be divisible by a, b, and by c
mighty shuttle
Have you heard of contrapositive @sonic tusk
sonic tusk
Prove that a/b or b/c or c/a is not an integer, the a/b+b/c+c/a is not an intege...
hmmmmmm
pastel vault
like if x, y, z are not all integers
sonic tusk
yes
pastel vault
x = a/b, y = b/c, z = c/a
sonic tusk
so if a/b, b/c, c/a are not integers
then their sum is also not an integer
by extension does that prove the statement itself?
sonic tusk
Prove that a/b or b/c or c/a is not an integer, the a/b+b/c+c/a is not an intege...
i feel like this is the best approach
pastel vault
so if a/b, b/c, c/a are not integers
you have to be careful
the correct negation is that x, y, z are not all integers
or at least one of x, y, z is not an integer
but yes, if you have proven the contrapositive of a statement
then the original statement must also be true
sonic tusk
the correct negation is that x, y, z are not all integers
or at least one of x,...
ohh ok
sonic tusk
then the original statement must also be true
alr
but how do i even go about proving that
it just makes intuitive sense since addition between integers will always be an integer
dusky cape
2x^2 + 12x + 23. Prove that it is >0
cedar kilnBOT
2x^2 + 12x + 23. Prove that it is >0
dusky cape
$2x^2 + 12x + 23$
wraith daggerBOT
Simon James B
dusky cape
so we can complete the (a+b)^2 formula but our a is sqrt2x right but in our 2ab we should have 2*sqrt2xb but there is no sqrt in our 2ab
idle gulch
so we can complete the (a+b)^2 formula but our a is sqrt2x right but in our 2a...
You can take that 2 common
dusky cape
$2(x^2 + 6x + 11) + 1$
wraith daggerBOT
Simon James B
dusky cape
is this it?
versed meadow
$2x^2+12x+23 > 2x^2 +2\times 3\sqrt{2} \times \sqrt{2}x + 18=(\sqrt{2}x+3\sqrt{2})^2$
dusky cape
why + 25
dusky cape
wht - 18
wraith daggerBOT
TimourX
versed meadow
ok now it's correct
pastel vault
2x^2 + 12x + 23. Prove that it is >0
alternative approach, verify that discriminant < 0
pastel vault
so this has no real roots (does not cross the x-axis)
and hence must lie entirely above, or entirely below, the x-axis
you can just test x = 0 then
idle gulch
alternative approach, verify that discriminant < 0
I'm sure he hasn't learned about the curves
dusky cape
I'm sure he hasn't learned about the curves
exactly
I am so so confused of what we have done lol
idle gulch
(square) + constant = positive for real x
pastel vault
I'm sure he hasn't learned about the curves
ah unfortunate
dusky cape
i am just at factoring chapter you know
pastel vault
handy method if you know discriminants, but yeah
idle gulch
Because the least value of a square term can be 0 for real x
pastel vault
there's always time to learn it
pastel vault
Because the least value of a square term can be 0 for real x
THIS
so many people do not know this
dusky cape
**Simon James B**
Sadly i did not understand anything. To many methods at the same time and for me they were to advanced. I understood all until this point. So we factored out 2 right? now our a is x and b is 3 so we can add 9 and substract 9. $2(x^2 + 6x + 9 -9 + 11) +1 => 2[(x+3)^2 +2]+1$
wraith daggerBOT
Simon James B
dusky cape
$2(x+3^2) +3 > 0$
wraith daggerBOT
Simon James B
dusky cape
seems that something went wrong. it should be +5 now +3 but i have no idea why
Nevermind i see it now let's take a step back
$2[(x+3)^2 +2] +1 => 2(x+3)^2 + 4 +1 => 2(x+3)^2 +5 > 0$
wraith daggerBOT
Simon James B
cedar kilnBOT
cedar kilnBOT
Available help channel!
sinful zephyr
If ab = 100
Bc = 200
Ca = 300
Then a + b + c = ?
cedar kilnBOT
If ab = 100
Bc = 200
Ca = 300
Then a + b + c = ?
sinful zephyr
What i tried =
ab + bc + ca = 600
600 ÷ 2 = 300
sinful zephyr
What i tried =
ab + bc + ca = 600
600 ÷ 2 = 300
Can we rewrite ab + bc+ ca as 2a + 2b + 2c?
frigid dust
If ab = 100
Bc = 200
Ca = 300
Then a + b + c = ?
cant you simply find a b and c here? there are 3 equations and 3 unknowns, should be solvable
sinful zephyr
cant you simply find a b and c here? there are 3 equations and 3 unknowns, shoul...
I amnt very smart
gritty viper
hint: find abc
sinful zephyr
thus 100/b=300/c
Well 300b = 100c?
frigid dust
Well 300b = 100c?
yes, and given that b*c=200, how can you find b and c?
sinful zephyr
Hmmm
frigid dust
hint: write both as b=....
sinful zephyr
and the other one?
Other one?
frigid dust
yes, and given that b*c=200, how can you find b and c?
this one is the other one
b*c=200
sinful zephyr
Btw is
2(a + b + c) will equal to ab + bc + ca?
It worked last time
frigid dust
Btw is
2(a + b + c) will equal to ab + bc + ca?
no
frigid dust
It worked last time
just because it works for some a,b,c doenst mean it works for all a,b,c
sinful zephyr
just because it works for some a,b,c doenst mean it works for all a,b,c
Dang why
frigid dust
because if we have two equations with b=, then we can set them equalt o each other
sinful zephyr
Well it wont ig
frigid dust
it would be fun to explore for what a,b and c: 2(a+b+c)=ab+bc+ac
sinful zephyr
A + b + c = 6
Ab = 4
Bc = 4
Ca = 4
frigid dust
but thats not the question, lets go back to the original question
$ab=100\bc=200\ac=300$
wraith daggerBOT
Bonk
sinful zephyr
Yeah
frigid dust
so, we have
\begin{equation}ab=100\end{equation}
\begin{equation}ac=300\end{equation}
wraith daggerBOT
Bonk
frigid dust
rewrite (1) as a=???
fallen heath
**Bonk**
Take product of equations (1) and (2) to get: $(ab)c^2 = 100 \times 300$ and you already know what ab is.
wraith daggerBOT
Arya
frigid dust
A = 100b
not quite
\begin{align*}ab&=100\\frac{ab}{b}&=\frac{100}{b}\a&=\frac{100}{b}\end{align*}
wraith daggerBOT
Bonk
sinful zephyr
Ac = 300
C = 300/a