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2003 AIME II problem 5 minor correction | Math for fun and glory | Khan Academy

Video vocabulary

anyway

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US / ˈeniweɪ /

UK / ˈɛniˌwe /

A1
  • adverb
  • A word used to change the topic of a conversation

shade

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US / ʃeɪd /

UK / ʃed /

A2
  • noun
  • Area of darkness when something blocks the light
  • Something that blocks or reduces light
  • Color that is a little darker or brighter than
  • verb
  • To shelter something from direct light
  • To change the truth to deceive or influence
  • To make one part of a picture darker or lighter

just

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US / dʒʌst /

UK / dʒʌst /

A1
  • adjecitve
  • Proper or appropriate; as is deserved
  • Doing or being what is right or fair
  • Legally correct; having enough evidence
  • adverb
  • Almost; nearly
  • Exactly at the time of
  • Exactly (the same as, like)

perpendicular

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US / ˌpɜ:pənˈdɪkjələ(r) /

UK / ˌpə:pənˈdɪkjəlɚ /

C1
  • adjecitve
  • Being at right angles to a given line or plane

clarify

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US / 'klærəfaɪ /

UK / ˈklærəˌfaɪ /

B1
  • verb
  • To make a liquid become clear or pure
  • To state again so that a point is clearer

for

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US / fə(r) /

UK / fɔr,fə /

A1
  • preposition
  • Used to show the purpose, or need of something

height

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US / haɪt /

UK / haɪt /

A2
  • noun
  • Distance of something from the bottom to the top

trick

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US / trɪk /

UK / trɪk /

A2
  • adjecitve
  • Not reliable; likely to deceive you
  • noun
  • Act of trying to fool someone
  • Quick or skillful way of doing something
  • Something done to make someone laugh
  • verb
  • To fool someone in order to obtain a result
  • To playfully tease or fool to make someone laugh

where

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US / hwɛə /

UK / hwɛr, wɛr /

A1
  • conjunction
  • At or in the place which
  • In whatever place
  • pronoun
  • Word you use to ask the location of something

theta

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US / ˈθi:tə /

UK / ˈθetə, ˈθi- /

C1
  • noun
  • 8th letter of the Greek alphabet

call

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US / kɔ:l /

UK / kɔl /

A1
  • noun
  • A order or request for action
  • The sound an animal makes, often when in danger
  • A correct prediction, e.g. a decision in a game
  • A short visit to a place or person
  • A decision by a referee or umpire in a game
  • verb
  • To make a request or order for action
  • (Animal) to make sounds to warn of danger
  • To announce the date of an election
  • To make a judgment about something
  • To give someone or something a name
  • To guess in advance the result or outcome
  • To visit a place or person for a short time
  • To phone someone
  • To speak, announce or shout in a loud voice

version

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US / ˈvə:ʃən /

UK / ˈvɚʒən, -ʃən /

A2
  • noun
  • Different way that someone interprets something
  • New or different form of something
  • An updated form of some software

stack

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US / stæk /

UK / stæk /

B1
  • noun
  • A large number or amount of something
  • Large tube through which smoke exits a building
  • verb
  • To arrange cards in a certain order, to cheat
  • To put one thing on top of another, e.g. boxes

incorrect

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US / ˌɪnkəˈrekt /

UK / ˌɪnkəˈrɛkt /

B1
  • adjecitve
  • Being wrong or inaccurate

hopefully

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US / ˈhəʊpfəli /

UK / ˈhopfəli /

A2
  • adverb
  • In a manner wishing something to happen

and

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US / ənd /

UK / ənd, ən,ænd /

A1
  • conjunction
  • Used to refer to two or more things
  • Plus; in addition; on top of that
  • Used to introduce an action that follows another

hey

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US / heɪ /

UK / he /

A1
  • exclamation
  • What you say when you want someone's attention

another

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US / əˈnʌðə(r) /

UK / əˈnʌðɚ /

A1
  • adjecitve
  • One more, but not this
  • determiner
  • One more added
  • pronoun
  • One more (thing)

over

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US / 'əʊvə(r) /

UK / ˈovɚ /

A1
  • adverb
  • Finished; no longer happening
  • Towards the ground/away from a standing position
  • noun
  • A set of six consecutive balls bowled in cricket
  • other
  • Finished
  • preposition
  • Showing the relation when a thing is above another
  • On or covering something
  • Above or across something
  • More than you need
  • From one side to the other
  • In the direction of; toward

out

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US / aʊt /

UK / aʊt /

A1
  • adverb
  • No longer stylish (e.g. fashion)
  • Away from home, business, or the usual place
  • In a direction/movement away from inside or center
  • noun
  • Act when a batter cannot continue playing
  • verb
  • To tell people a secret

minus

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US / ˈmaɪnəs /

UK / ˈmaɪnəs /

A2
  • adjecitve
  • Being close to the bottom of a letter grade
  • noun
  • Negative quality; disadvantage
  • Sign that indicates subtraction
  • preposition
  • Being less than zero in number
  • Used when subtracting one number from another
  • verb
  • To take one number away from another; subtract

along

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US / əˈlɒŋ /

UK / əˈlɔ:ŋ /

A1
  • adverb
  • As a companion; taken with someone
  • preposition
  • At a point on a line

anything

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US / ˈeniθɪŋ /

UK / ˈɛniˌθɪŋ /

A1
  • pronoun
  • A thing of any kind

aim

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US / eɪm /

UK / em /

A2
  • noun
  • Act of pointing something at a target
  • Purpose or goal
  • verb
  • To point something at a target
  • To intend to get a thing or for a thing to happen

obviously

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US / ˈɒbviəsli /

UK / ˈɑːbviəsli /

A2
  • adverb
  • In a way that is obvious/easy to see or understand

matter

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US / 'mætə(r) /

UK / ˈmætɚ /

A1
  • noun
  • Material all things are made of that fills space
  • Problem or reason for concern
  • verb
  • To be of great importance; to count

ourselves

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US / ɑ:ˈselvz /

UK / aʊrˈsɛlvz, ɑr- /

A2
  • pronoun
  • (Showing speaker is affected by their own action)

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

A1
  • adverb
  • Used to add new (often different) information

quick

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US / kwɪk /

UK / kwɪk /

A1
  • adjecitve
  • Using very little time; moving far in little time
  • Learning and understanding rapidly and easily

here

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US / hɪə(r) /

UK / hɪr /

A1
  • adverb
  • In this place; at this location, position or time

axis

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US / ˈæksɪs /

UK / ˈæksɪs /

B1
  • noun
  • A line about which a rotating body turns

side

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US / saɪd /

UK / saɪd /

A1
  • adjecitve
  • In addition to the central or most important thing
  • Next to, or very close to, the right or left of
  • noun
  • Position or opinion that is opposite to another
  • Right or left part of a person's body
  • Feature of someone's personality
  • Straight edge of an object
  • One of two or more groups fighting each other
  • verb
  • To decide to agree with one point, not the other

plus

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US / plʌs /

UK / plʌs /

A2
  • noun
  • A benefit or advantage
  • preposition
  • In addition; add
  • In addition to; Indicates a number above
  • Having a value greater than zero

take

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US / teɪk /

UK / tek /

A1
  • noun
  • Filming of a single scene for TV or the movies
  • verb
  • To accept something that is offered
  • To use a method of traveling, e.g. a bus or train
  • To use a camera to make (a photo)
  • To pick up something and go away with it
  • To study a subject in school
  • To attempt to pass (an exam)
  • To require a certain amount of time, money, space

add

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US / æd /

UK / æd /

A1
  • verb
  • To include as well
  • To put two things or numbers together, e.g. 2+2

log

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US / lɒg /

UK / lɔ:g /

B1
  • noun
  • Detailed account of events and progress on journey
  • Piece of a tree that has fallen or been cut
  • verb
  • To move a certain distance or speed, as recorded
  • To leave a website or computer account
  • To record something in an official account
  • To cut trees down and take the wood to use it

clearly

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US / ˈklɪəlɪ /

UK / ˈklɪrlɪ /

A1
  • adverb
  • In a way that is easy to understand; obviously

look

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US / lʊk /

UK / lʊk /

A1
  • noun
  • Person's beauty or physical attractiveness
  • Particular style of dress, clothes, or appearance
  • The way someone, something appears to be
  • Act of searching for or examining something
  • verb
  • To appear to be; seem

show

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US / ʃəʊ /

UK / ʃo /

A1
  • noun
  • False display of love, emotion or action
  • An event for displaying or promoting goods
  • Play, opera or theater production
  • Presentation that is broadcast on TV or radio
  • verb
  • To be easily seen or displayed
  • To display your emotions or feelings
  • To let someone see something; display
  • To explain or teach how something is done
  • To lead someone somewhere; point them to something
  • To prove something to be true, correct
  • To reveal or allow something to be seen

way

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US / weɪ /

UK / we /

A1
  • adverb
  • Used to emphasize the amount of something
  • noun
  • Particular aspect of something being thought about
  • One of various choices that could be made
  • Particular existing condition or state
  • (Often long) distance or length of time
  • How something is done
  • Route to go to a place
  • Small street
  • Usual behavior, emotions or habits of someone

figure

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US / ˈfiɡə /

UK / ˈfɪɡjɚ /

A1
  • noun
  • Your body shape
  • Numbers in a calculation
  • Doll-like thing meant to represent a person
  • Picture or diagram giving information in a text
  • Person who is very important or famous
  • Shape of a person seen indistinctly or in outline
  • Amount that is expressed in numbers
  • verb
  • To appear in a game, play or event
  • To calculate how much something will cost
  • To understand after thinking; work out

work

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US / wɜ:k /

UK / wɚk /

A1
  • adjecitve
  • Activity completed in order to make money
  • noun
  • The product of some artistic or literary endeavor
  • Everything created by an author, artist, musician
  • Activity you do in order to make money
  • Place of employment
  • Final product or result of a some labor or effort
  • verb
  • To bring into a specific state of success
  • To be functioning properly, e.g. a car
  • To do your job in your company or workplace
  • To make an effort to do something
  • To operate a machine or device

top

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US / tɒp /

UK / tɑp /

A1
  • adjecitve
  • Being good or great; excellent
  • Being the most in degree or quality
  • noun
  • Item of clothing covering shoulders to waist, hips
  • A cover or lid
  • Part (e.g. road) that is the furthest away
  • Highest or upper part or point of something
  • Best or highest status or position
  • Flat upper surface or edge of an object
  • Child's toy that spins on its point
  • Upper section or leaves of fruits and vegetables
  • Winning in a sport or contest
  • verb
  • To be higher or larger than a specific value
  • To arrive at the highest point of something
  • To put a cover or layer or over something
  • To kill; murder
  • To hold the highest position on a list
  • To be more important, successful or better than
  • To beat someone or win at a game

apologize

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US / ə'pɒlədʒaɪz /

UK / əˈpɑləˌdʒaɪz /

A2
  • verb
  • To say sorry because of a mistake or injury

very

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US / 'verɪ /

UK / ˈvɛri /

A1
  • adjecitve
  • Concerning a large degree or amount
  • adverb
  • A lot; much; many
  • Used to stress or emphasize a noun

volume

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US / ˈvɒlju:m /

UK / ˈvɑljum, -jəm /

A2
  • noun
  • Number or amount of something e.g. sales
  • Amount of space that something takes up
  • One book that is part of a set of books
  • Level of sound produced by radio or television

they

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US / ðeɪ /

UK / ðe /

A1
  • pronoun
  • Two or more people, animals, or things

incorrectly

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US / ˌɪnkə'rektlɪ /

UK / ˌɪnkə'rektlɪ /

B1
  • adverb
  • In a way that is incorrect or wrong

cylinder

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US / ˈsɪlɪndə(r) /

UK / ˈsɪləndɚ /

B2
  • noun
  • Tube shape with flat ends

solve

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US / sɒlv /

UK / sɑ:lv /

A2
  • verb
  • To figure out a way to handle a difficult problem

right

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US / raɪt /

UK / raɪt /

A1
  • adjecitve
  • Correct or true
  • The south side of you looking at the rising sun
  • adverb
  • Exactly in a place or time
  • Being appropriate for a particular event
  • In a straight or direct manner
  • noun
  • Something you are legally or morally allowed to do
  • Ideal of what is just and good
  • Political view based on conservatism
  • verb
  • To fix something previously wrong or unjust
  • To make upright, as a boat that has turned over

then

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US / ðen /

UK / ðɛn /

A1
  • adverb
  • A time that is not now
  • Following another thing in time or arrangement
  • Following on reasonably from what was just stated
  • At that time, not now

remind

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US / rɪ'maɪnd /

UK / rɪˈmaɪnd /

A2
  • verb
  • To cause someone to remember what they forgot

what

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US / wɒt /

UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /

A1
  • pronoun
  • Thing or things that; the same kind as
  • Used to emphasize something you are about to say
  • adjecitve
  • Used to refer to a particular amount, or number
  • Used to place emphasis on a particular quality
  • adverb
  • To a certain degree
  • determiner
  • Question words used for asking for information
  • exclamation
  • Used to express excitement, shock, or surprise
  • Used to ask people to repeat something they said

that

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US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
  • The person or thing being talked about
  • Used to identify something both the speakers know
  • adverb
  • Used to reinforce adjectives and adverbs
  • To such a degree
  • conjunction
  • Used to connect a noun clause
  • Used to introduce an adverbial clause
  • Used to introduce a noun clause
  • determiner
  • Used as a noun to refer to something
  • pronoun
  • Referring an object far away
  • Used to refer to the relative pronoun ‘which’

like

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US / laɪk /

UK / laɪk /

A1
  • adjecitve
  • Being equal in status or kind to something else
  • Similar; the same
  • adverb
  • For example
  • preposition
  • Used as a filler; used to ad emphasis
  • In a way similar to the way something is done
  • Such as; for example
  • verb
  • To want to
  • To find something pleasing; to prefer something

end

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US / end /

UK / ɛnd /

A1
  • noun
  • The point where a process or activity stops
  • A point that marks the limit of something; finish
  • Goal or desired result
  • The tip, edge or limit of something
  • verb
  • To be the last part of something

see

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US / si: /

UK / si /

A1
  • verb
  • To go on a date with someone
  • To predict whether something is, or will be true
  • To understand what someone says
  • To use your eyes to look at something
  • To visit a place often for sightseeing or pleasure
  • To visit or meet with someone
  • To find out by waiting or looking
  • To watch (a game, movie or TV show)

misspeak

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US / ˌmɪs'spi:k /

UK / mɪs'spi:k /

A1
  • intransitive verb
  • Express oneself insufficiently clearly or accurately.

properly

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US / ˈprɔpəlɪ /

UK / ˈprɑːpərli /

A2
  • adverb
  • In an appropriate or correct manner

know

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US / nəʊ /

UK / noʊ /

A1
  • verb
  • To be familiar with a person or place, thing
  • To feel sure or be convinced about something
  • To discover or be aware of something
  • To be able to distinguish
  • To have knowledge of things

could

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US / kəd /

UK / kʊd /

A1
  • other
  • Past form of 'can' to mean have ability
  • Used to ask for something politely, or to offer

last

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US / lɑ:st /

UK / læst /

A1
  • adjecitve
  • Most recent; the latest
  • The one before
  • adverb
  • Final; with nothing following; after all others
  • Most recent; the latest
  • determiner
  • The one at the end
  • verb
  • To remain or continue for a certain period of time

have

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US / həv /

UK / hæv /

A1
  • verb
  • To drink, smoke, eat or use something
  • To experience the effects of something
  • To organize an event, or join in an activity
  • To own, possess, or hold something
  • To cause to happen or produce a particular effect

draw

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US / drɔ: /

UK / drɔ /

A1
  • noun
  • Something that attracts people to visit a place
  • A lottery or prize
  • Result of a game, contest where the score is equal
  • verb
  • To attract attention to someone or something
  • To get closer to or approach something or someone
  • To influence a person's involvement in something
  • To move an object by pulling
  • To create an image using pen or pencil and paper
  • To take one thing out of a container, etc.
  • (Of 2 teams) to finish a game with the same score

problem

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US / ˈprɒbləm /

UK / ˈprɑbləm /

A1
  • adjecitve
  • Causing trouble
  • noun
  • Something difficult to deal with or causes trouble
  • Question to show understanding of a math concept

cut

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US / kʌt /

UK / kʌt /

A1
  • noun
  • Act/instance of reducing the size or amount of
  • Small opening made using a sharp object, knife
  • A wound on someone's skin caused by a sharp object
  • verb
  • To reduce the size or amount of something
  • To use knife or scissors to divide, open surfaces

but

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US / bət /

UK / bʌt,bət /

A1
  • conjunction
  • Used before you say something different, opposite

video

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US / 'vɪdɪəʊ /

UK / ˈvɪdiˌo /

A2
  • adjecitve
  • Of a moving film, with sound
  • noun
  • Recording capturing action with sound
  • verb
  • To make a moving film of, with sound

want

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US / wɔnt /

UK / wɑnt, wɔnt /

A1
  • noun
  • Something you desire or wish for
  • State or condition of poverty
  • verb
  • To desire or wish for something; hope for a thing

would

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US / wʊd /

UK / wʊd /

A1
  • other
  • Used to indicate something imagined
  • Used to ask or request something politely
  • Verb used to indicate possibility, request etc.
  • Used to indicate a preference
  • Used to indicate a reaction
  • Indicates something that occurred repeatedly
  • Used to indicate willingness or ability to do

can

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US / kæn /

UK / kən,kæn /

A1
  • noun
  • A metal container for storing food
  • other
  • Able to; have the skill to; have the time to
  • verb
  • To have permission to
  • To be possible to do
  • To preserve food by sealing it in a metal container

now

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US / naʊ /

UK / naʊ /

A1
  • adverb
  • At the present time or moment
  • Right away or in the next possible moment
  • Used to refer to something that just happened
  • Time between the present and a moment in the past
  • conjunction
  • Resulting from or because of something

angle

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US / ˈæŋgl /

UK / ˈæŋɡəl /

B1
  • noun
  • Place where two lines or surfaces meet

reasonable

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US / ˈri:znəbl /

UK / ˈrizənəbəl /

A2
  • adjecitve
  • Being somewhat good, but not great
  • Fair and sensible; appropriate

not

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US / nɒt /

UK / nɑt /

A1
  • adverb
  • Word indicating the negative
  • (Used to form the negative of verbs)

visualize

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US / ˈvɪʒuəlaɪz /

UK / ˈvɪʒuəˌlaɪz /

B1
  • verb
  • To create a picture in your mind of something

round

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US / raʊnd /

UK / raʊnd /

A1
  • adjecitve
  • Being in the shape of a circle or ball
  • Consisting of curved and not straight lines
  • Somewhat fat or curved in physical appearance
  • Being a whole number, not a fraction, decimal
  • adverb
  • From beginning to end (e.g. of the year)
  • noun
  • One of the three-minute periods in boxing
  • Drinks bought by one person for a group
  • Series regular visits, routes, or stops at places
  • Single shot or bullet fired from a gun
  • An object with a circular shape
  • One stage in a series of games, e.g. football
  • One's turn in a game such as cards
  • preposition
  • Going around
  • verb
  • To cause or make a circular or curved shape
  • To go or pass around the bend or corner
  • To change from a fraction to nearest whole number

same

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US / seɪm /

UK / sem /

A1
  • adjecitve
  • Being equal in quality and amount
  • Identical; not different
  • Not changing; constant

make

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US / meɪk /

UK / mek /

A1
  • noun
  • Product made by a particular company; brand
  • verb
  • To arrange or prepare something e.g. dinner
  • To get to (a place) on time
  • To gain the status of (teacher, lawyer, etc.)
  • To create something by putting things together
  • To earn a certain amount of money at a job
  • To cause or force a person to do something
  • To cause something to happen or be formed

than

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US / ðən /

UK / ðən /

A1
  • conjunction
  • Used to compare two things that are not equal

one

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US / wʌn /

UK / wʌn /

A1
  • pronoun
  • You; we; people in general; someone
  • The thing or person previously referred to
  • adjecitve
  • Used to refer to people generally
  • Being a strong example of (something mentioned)
  • number
  • Number 1

lucky

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US / 'lʌkɪ /

UK / ˈlʌki /

A1
  • adjecitve
  • Fortunate; having good things happen by chance

you

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US / jʊ /

UK / ju /

A1
  • pronoun
  • Person someone is speaking or writing to
  • Person or people in general

kind

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US / kaɪnd /

UK / kaɪnd /

A1
  • adjecitve
  • In a caring and helpful manner
  • friendly and considerate
  • noun
  • One type of thing

also

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US / 'ɔ:lsəʊ /

UK / ˈɔlso /

A1
  • adverb
  • In addition; too; in a similar way

this

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US / ðɪs /

UK / ðɪs /

A1
  • adjecitve
  • Used to indicate something already been discussed
  • Used to indicate something happening around now
  • adverb
  • An indication by a physical motion or gesture
  • A specific amount of
  • determiner
  • Person, thing, or idea near you
  • pronoun
  • Person, thing that has already been discussed

flip

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US / flɪp /

UK / flɪp /

B2
  • noun
  • Act of turning your body in the air; somersault
  • Movement of something from one position to another
  • verb
  • To turn your body in the air, as in gymnastics
  • To move into a different position quickly
  • To move a button or switch to turn on or off

visualization

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US / ˌvɪʒʊəlaɪ'zeɪʃn /

UK / ˌvɪʒʊəlɪˈzeʃən /

C1
  • noun
  • Act of creating an image in your mind

the

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US / ðə /

UK / ðə /

A1
  • article
  • Used to refer to something already mentioned
  • Used to show there is only one of something
  • determiner
  • Used to describe all of a family

let

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US / let /

UK / lɛt /

A1
  • verb
  • To allow someone to do something
  • Introducing a suggestion to do something together
  • To rent a house, etc. to others

big

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US / bɪɡ /

UK / bɪɡ /

A1
  • adjecitve
  • Popular
  • Serious
  • Large

solution

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US / səˈlu:ʃn /

UK / səˈluʃən /

A2
  • noun
  • Mix of a liquid and a solid or a gas
  • Way to solve or deal with a problem

less

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US / les /

UK / lɛs /

A1
  • adjecitve
  • Smaller in amount or degree
  • adverb
  • To a smaller degree
  • noun
  • A smaller amount
  • preposition
  • With the subtraction of
  • Subtitles section
  • I want to do a quick correction to the AIM problem, where we cut out a wedge of a log.
  • In that problem, so we had a wedge.
  • So just to remind ourselves, we had a log that looked like that.
  • We made one cut that was right along, that was perpendicular to the axis of the log.
  • We made another cut that had a 45 degree angle.
  • So it was like that.
  • And the solution to the problem, we had to figure out the volume of the wedge.
  • And the big trick there was, so if I drew the wedge like this, so let me take a side view of the wedge.
  • So this is a side of the wedge right over here.
  • I could shade it to show that it's kind of round.
  • So if I were to shade it, you can kind of see that it is round.
  • So this is a side view of the wedge.
  • We knew that this right here is a 45 degree angle.
  • And the way that we solved the problem, or the kind of trick, was to say, hey, if we had another wedge like this, and we just stacked it on top of this one, if we just flipped it over and made it like this,
  • Then we also have a 45 degree angle over here.
  • And then we would make a cylinder.
  • And it's very easy to figure out the height of a cylinder.
  • And I misspoke, and I said it was lucky that we had a 45 degree angle.
  • Because in the last video, I was actually visualizing it incorrectly.
  • I said, oh, because if you had anything less than a 45 degree angle, this wouldn't have worked out properly.
  • But I was wrong.
  • It would work out properly.
  • Let me show you.
  • So if this, let's say, whatever angle you have here.
  • So this is going to be 90 degrees over here.
  • Whatever angle you have over here.
  • So let's call that theta.
  • This over here, they have to add up to 180.
  • So theta plus 90 plus this has to be 180.
  • Or theta plus this has to be 90. Or this could be, you could just call this 90 minus theta.
  • Now, if I were to take the same thing, if I took another kind of wedge like this and flipped it over, over this thing, it's not this angle that's going to be right here.
  • It's this angle.
  • So let me just draw it right over here.
  • So it would look like this.
  • It would look like this, where this now is theta.
  • This end right here is actually that end on the flipped version.
  • And now this angle right over here is now 90 minus theta.
  • So clearly, you have 90 minus theta plus theta.
  • So this thing is going to be a right angle no matter what.
  • Obviously, it has to be a reasonable angle.
  • But no matter, assuming that it's less than 90 degrees, it would have worked out.
  • You could have done the same trick.
  • Anyway, hopefully that clarifies it.
  • And I apologize for the incorrect visualization in the last video.