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Analyzing related rates problems: equations (trig) | AP Calculus AB | Khan Academy

Video vocabulary

three

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US / θri: /

UK / θri /

A1
  • number
  • Number 3

wall

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US / wɔ:l /

UK / wɔl /

A1
  • noun
  • High, flat side of a room or building

able

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US / 'eɪbl /

UK / ˈebəl /

A2
  • adjecitve
  • Skilled; good at something
  • Having the ability to do something

you

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US / jʊ /

UK / ju /

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

little

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US / 'lɪtl /

UK / ˈlɪtl /

A1
  • adjecitve
  • Being short in length of time
  • Not great in size, amount, or degree; small
  • Being very young or younger than others
  • adverb
  • By an amount or degree that is not very much
  • Not often; rarely
  • noun
  • A person's name

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

how

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US / haʊ /

UK / haʊ /

A1
  • adjecitve
  • To what degree? in what amount?
  • adverb
  • (Used to explain the way to do something)
  • exclamation
  • For what purpose or reason?
  • (Used to ask someone or something's condition)

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

hypotenuse

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US / haɪˈpɒtənju:z /

UK / haɪˈpɑtn:ˌus, -ˌjus /

C2
  • noun
  • Side of a right triangle opposite the right angle

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

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

order

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US / 'ɔ:də(r) /

UK / ˈɔrdɚ /

A1
  • noun
  • Religious or social group, as of nuns
  • Instructions to others to do what you want
  • State in which things are neat and organized
  • What you ask to buy, e.g. food in restaurant
  • Social or political control of an area
  • verb
  • To put things in a sequence, e.g. a, b, c
  • To give instructions to others to do what you want
  • To ask for something you want to buy, e.g. a meal

might

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US / maɪt /

UK / maɪt /

A1
  • other
  • Used to politely request an action
  • Used to express that something could happen
  • Power, strength, force or influence

also

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US / 'ɔ:lsəʊ /

UK / ˈɔlso /

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

other

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US / 'ʌðə(r) /

UK / ˈʌðər /

A1
  • adjecitve
  • A thing different from that mentioned
  • determiner
  • (Something) else; not the first (one)
  • pronoun
  • Being the one and only thing of that category

rate

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US / reɪt /

UK / ret /

A1
  • noun
  • Current price or cost of something
  • Speed or frequency of events over time
  • verb
  • To assess something or consider its qualities

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

opposite

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US / ˈɔpəzit /

UK / ˈɑpəzɪt /

A2
  • adjecitve
  • Being totally different from
  • noun
  • Person, thing that is totally different to another
  • preposition
  • Across from or on the side facing 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

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

instant

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US / ˈɪnstənt /

UK / ˈɪnstənt /

A2
  • adjecitve
  • (Food) requiring very little preparation
  • Occurring immediately
  • noun
  • A very short period of time

sine

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US / saɪn /

UK / saɪn /

C1
  • noun
  • Angle's ratio of side opposite hypotenuse

with

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US / wɪð /

UK / wɪð, wɪθ /

A1
  • preposition
  • As an employee of a certain company
  • Against something or someone
  • By using something as a means
  • Due to; because of
  • Including
  • Possessing as a feature
  • Related to
  • (Of the manner/attitude by which you do something)
  • Used to express that people or things are together

negative

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US / 'neɡətɪv /

UK / ˈnɛɡətɪv /

A2
  • adjecitve
  • Being harmful, unwanted or unhelpful
  • In mathematics, being less than zero
  • Focusing on the bad aspects; pessimistic
  • noun
  • The opposite to a positive electrical charge
  • In grammar, containing words such as 'no' or 'not'
  • Reply to a question or statement that means 'no'
  • Image on camera film used to make a photo

clear

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US / klɪə(r) /

UK / klɪr /

A1
  • adjecitve
  • Empty; without anything in it or on it
  • Being certain and without doubt
  • Being without any streaks, spots, or cracks
  • Easy to understand; well-explained; obvious
  • adverb
  • Completely; entirely; through the whole of
  • Away from; free from
  • verb
  • To recover from or remove a problem
  • To get rid of the entire contents of
  • To give permission for something to happen
  • To make a place or surface empty by removing
  • To prove that someone is not guilty or to blame

moment

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US / 'məʊmənt /

UK / 'moʊmənt /

A1
  • noun
  • Very short or brief period of time
  • A particular period in time

unit

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US / ˈju:nɪt /

UK / ˈjunɪt /

B2
  • noun
  • One apartment in a building
  • Group of people, e.g. soldiers, that work together
  • A chapter of a textbook
  • A standard of measurement of something
  • Single item made by a company

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

increase

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US / ɪn'kri:s /

UK / ɪnˈkris /

A1
  • noun
  • Fact of increasing; amount something grows by
  • verb
  • To make or become larger in size or amount

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

meter

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US / ˈmi:tə(r) /

UK / ˈmitɚ /

A1
  • noun
  • Unit of length equal to 100 cm or about 39 inches
  • Device for measuring things e.g. gas
  • Device to put money into to rent a parking place
  • Rhythm of a poem
  • verb
  • To measure water, etc. when it has to be paid for

prime

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US / praɪm /

UK / praɪm /

A2
  • adjecitve
  • Concerning the highest in quality or value
  • Concerning the highest level of achievement
  • Most important, chief importance in value
  • verb
  • To get a person ready for; prepare someone
  • To coat with paint to prepare it for the main coat

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

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)

information

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US / ˌɪnfəˈmeɪʃn /

UK / ˌɪnfɚˈmeʃən /

A1
  • noun
  • Collection of facts and details about something

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

solve

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US / sɒlv /

UK / sɑ:lv /

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

tell

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US / tel /

UK / tɛl /

A1
  • verb
  • To (strongly) advise a person to do something
  • To be able to guess
  • To say or communicate information to someone
  • To inform, but without speaking; indicate
  • To have a negative effect on someone
  • To order someone to do something; command
  • To recognize something based on certain signs
  • To report something to an authority
  • To explain something or teach someone something

ladder

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US / 'lædə(r) /

UK / ˈlædɚ /

B1
  • noun
  • Series of levels to go up in society or business
  • Device with a series of steps used for climbing

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

interesting

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US / ˈɪntrəstɪŋ /

UK / ˈɪntrɪstɪŋ, -tərɪstɪŋ, -təˌrɛstɪŋ /

A1
  • adjecitve
  • Taking your attention; making you want to know
  • verb
  • To make someone want to know about something
  • To persuade to do, become involved with something

ground

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US / graʊnd /

UK / ɡraʊnd /

A1
  • noun
  • Connection of electrical equipment to earth
  • Place where people play particular sports
  • Solid surface of the earth that we stand on
  • verb
  • To break (coffee, etc.) into tiny bits with machine
  • To make loud jarring noise by pressing hard
  • To make sharp or smooth through friction
  • To press together with force making a sound
  • To punish someone by not allowing them to go out
  • To hit the bottom
  • (Of boat) to hit the bottom so it cannot move
  • To prevent a plane from taking off
  • To place the ball over the goal line in rugby

get

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US / ɡet /

UK / ɡɛt /

A1
  • verb
  • To become affected by illness or disease
  • To begin to feel or understand an emotion or idea
  • To send or bring someone or something home, etc.
  • To go somewhere to obtain something
  • To (cause to) do a particular thing
  • To obtain, receive or be given something
  • To currently have
  • To prepare for use
  • To understand something being said or read

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

find

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US / faɪnd /

UK / faɪnd /

A1
  • noun
  • Object discovered by chance
  • Things that have been discovered from searching
  • verb
  • To become aware of something that is happening
  • To discover or meet by chance
  • To decide if someone is guilty of a crime, or not
  • To learn through study, research, or effort
  • To discover something by looking for it

happen

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US / 'hæpən /

UK / ˈhæpən /

A1
  • verb
  • To take place or occur

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

step

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US / step /

UK / stɛp /

A1
  • noun
  • Movement done as part of a particular dance
  • Distance covered by one movement of a leg; stride
  • One part or stage in a process
  • Sound made by the feet while walking; footstep
  • Flat horizontal piece that forms stairs
  • verb
  • To raise and moving the foot to put it down

value

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US / 'vælju: /

UK / ˈvælju /

A1
  • noun
  • Importance you attach to something
  • Some information expressed as a number
  • Stated price of something; worth
  • verb
  • To place a high importance on something
  • To say how much money something is worth

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’

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

care

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US / keə(r) /

UK / kɛr /

A1
  • noun
  • Treatment you receive for your health or safety
  • Something that a person worries about
  • verb
  • To feel interest, concern, or worry
  • To look after someone or something

respect

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US / rɪ'spekt /

UK / rɪˈspɛkt /

A2
  • noun
  • Behavior that follows the established rules
  • Admiration for someone
  • Regard or admiration for someone or something
  • verb
  • To follow the established rules
  • To think very highly of another person

down

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US / daʊn /

UK / daʊn /

A1
  • adjecitve
  • Being sad, sick, or unhappy
  • Not operating, working, or functioning as expected
  • adverb
  • In a sitting or lying position
  • Towards or on the ground, floor, or bottom
  • Written or copied onto a piece of paper or book
  • noun
  • A period of play in American Football
  • The soft, smooth hairs that cover a bird's body
  • preposition
  • Along something's path, course, or direction
  • Going from a higher position to a lower position
  • verb
  • To eat or drink very quickly and without stopping

scroll

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US / skrəʊl /

UK / skroʊl /

B2
  • noun
  • A long rolled piece of paper with writing on it
  • Something shaped like a rolled piece of paper
  • other
  • To move up, down, or across a computer screen
  • verb
  • To move up and down a computer screen

lean

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US / li:n /

UK / lin /

A2
  • adjecitve
  • Having a low fat content
  • Efficient; well-designed with no waste
  • Thin; slim; skinny; with little fat on your body
  • verb
  • To balance against or on something for support
  • To have a tendency to do something; favor

about

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US / ə'baʊt /

UK / əˈbaʊt /

A1
  • adverb
  • To be ready to; going to
  • Nearly; approximately; roughly
  • Near; next to; close to
  • In every direction; in the area of
  • preposition
  • Concerning or of (a topic)

bottom

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US / 'bɒtəm /

UK / ˈbɑtəm /

A2
  • adjecitve
  • Concerning the part under something, e.g. a vase
  • noun
  • The part of your body you sit on
  • Lowest position, often in a company or contest
  • Lowest part of something; part on which it rests
  • Lowest level, part or point of an object
  • The part under something, e.g. a vase
  • verb
  • To hit the lowest part of something

first

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US / fɜ:st /

UK / fɚst /

A1
  • adjecitve
  • 1st; Coming before all others in time or place
  • noun
  • Something that has not happened before
  • Position of the winner of a competition

here

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US / hɪə(r) /

UK / hɪr /

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

certain

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US / ˈsə:tn /

UK / ˈsɚtn /

A1
  • adjecitve
  • Definite, fixed
  • Referring to an unspecified amount or quantity
  • Being sure about something; without doubt

straight

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US / streɪt /

UK / stret /

A2
  • adjecitve
  • Not having curves, bends, or angles
  • Not gay; heterosexual
  • honest and direct
  • adverb
  • in a line; immediately; honestly and directly

try

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US / traɪ /

UK / traɪ /

A1
  • noun
  • Act of putting the ball over the rugby goal line
  • verb
  • To make an effort, to attempt to do something
  • To examine someone or something in a court of law
  • To test to see if it is good or you like it
  • To make a person go through hardship; test
  • To examine something to see if it works

basic

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US / 'beɪsɪk /

UK / ˈbesɪk /

A1
  • adjecitve
  • At the most important or easiest (beginner) level

will

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US / wɪl /

UK / wɪl /

A1
  • noun
  • Desire or choice of someone
  • Document saying who gets your money when you die
  • Strong desire to do something; determination
  • Desire to do something; strong wish to do
  • other
  • Used with verbs to express the future
  • verb
  • To leave property to someone after your death
  • To use mental effort to make something happen
  • Am (is, are) likely to do
  • To influence someone to do something
  • To make something happen by strongly wanting it

trigonometric

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US / ˌtrɪɡənə'metrɪk /

UK / ˌtrɪɡənə'metrɪk /

A1
  • other
  • Of or relating to or according to the principles of trigonometry

write

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US / raɪt /

UK / raɪt /

A1
  • verb
  • To compose letters and words on paper or a screen
  • To invent or create a computer program
  • To create a song or piece of music

ourselves

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US / ɑ:ˈselvz /

UK / aʊrˈsɛlvz, ɑr- /

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

either

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US / ˈaɪðə(r) /

UK / ˈiðɚ, ˈaɪðɚ /

A1
  • adjecitve
  • One of two (used when there is a choice of two)
  • adverb
  • (After negative statements) likewise, in addition
  • conjunction
  • Used as the first of two connected things
  • determiner
  • Used as the first of two connected things
  • pronoun
  • The one or the other

divide

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US / dɪ'vaɪd /

UK / dɪˈvaɪd /

A2
  • verb
  • To split numbers by another number, e.g. 6 / 2 = 3
  • To separate something into equal pieces

use

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US / ju:s /

UK / ju:s /

A1
  • noun
  • Ability to employ something
  • Need to enable it to work
  • Permission to do something with something
  • Act of spending money
  • Operation of a machine for its function
  • verb
  • To have; to eat, drink, consume etc.
  • To need to operate or work
  • To operate a machine, e.g. a computer
  • To treat a person badly
  • To do something with, for a task or purpose

relate

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US / rɪ'leɪt /

UK / rɪˈlet /

A2
  • verb
  • To demonstrate a logical relationship between
  • To understand or sympathize with, e.g. a problem
  • To tell a story, or tell about a particular event

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

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

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

always

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US / 'ɔ:lweɪz /

UK / ˈɔlwez,-wɪz,-wiz /

A1
  • adverb
  • For a very long time
  • At all times; in every situation
  • For ever

new

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US / nju: /

UK / nu, nju /

A1
  • adjecitve
  • Not old, recently born, built or made
  • Being or doing something for the first time
  • Not felt, seen or known previously
  • Having been recently bought or obtained

trigonometry

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US / ˌtrɪgəˈnɒmətri /

UK / ˌtrɪɡəˈnɑmɪtri /

A1
  • noun
  • Mathematics of triangles and trigonometric functions

theta

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US / ˈθi:tə /

UK / ˈθetə, ˈθi- /

C1
  • noun
  • 8th letter of the Greek alphabet

estate

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US / ɪˈsteɪt /

UK / ɪˈstet /

A2
  • noun
  • A large piece of land, usually with a large house
  • Group of houses all built at the same time

from

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US / frəm /

UK / frʌm, frɑm,frəm /

A1
  • preposition
  • Starting at a particular place, time or level

distance

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US / 'dɪstəns /

UK / ˈdɪstəns /

A2
  • noun
  • Lack of friendliness toward another person
  • Amount of space between two places or things
  • verb
  • To make it clear that you are not involved

something

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US / ˈsʌmθɪŋ /

UK / ˈsʌmθɪŋ /

A1
  • adverb
  • In some degree or measure
  • noun
  • a person or thing of some value or consequence
  • pronoun
  • Thing that is not yet known or named
  • Approximately

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

come

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US / kʌm /

UK / kʌm /

A1
  • verb
  • To arrive at a place
  • (Of mail) to be delivered
  • To move toward someone; go with someone
  • To reach a certain point or state

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

appropriate

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US / ə'prəʊprɪət /

UK / əˈpropriɪt /

A2
  • adjecitve
  • Right or suitable for some situation or purpose
  • verb
  • To set aside money for a specific later purpose
  • To take (money or goods) without permission

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

think

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US / θɪŋk /

UK / θɪŋk /

A1
  • verb
  • To have an idea about something without certainty
  • To have an idea, opinion or belief about something

real

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US / rɪəl /

UK / ˈriəl, ril /

A1
  • adjecitve
  • Actually existing or happening, not imagined
  • Being important or special

bit

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US / bɪt /

UK / bɪt /

A1
  • noun
  • Device put in a horse's mouth to control it
  • Small piece of something
  • verb
  • (E.g. of fish) to take bait and be caught

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

between

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US / bɪˈtwi:n /

UK / bɪˈtwin /

A1
  • preposition
  • From one person, thing, or place, to another
  • In the space that separates two objects
  • Among two or more people who share something

sub

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US / sʌb /

UK / sʌb /

B2
  • prefix
  • A prefix meaning 'under' or 'instead of'

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

all

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US / ɔ:l /

UK / ɔl /

A1
  • adverb
  • Completely; totally
  • determiner
  • Being every one of something
  • pronoun
  • 100% of something; the complete amount

useful

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US / ˈju:sfl /

UK / ˈjusfəl /

A1
  • adjecitve
  • That can help with a particular task

both

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US / bəuθ /

UK / boθ /

A1
  • determiner
  • Used to refer to two things at the same time
  • pronoun
  • Used to refer to two things at the same time

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

multiply

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US / ˈmʌltɪplaɪ /

UK / ˈmʌltəˌplaɪ /

B1
  • verb
  • To increase in number by reproducing/having babies
  • To increase something by a large quantity
  • To have increased by a large quantity
  • To calculate two numbers e.g. 2 x 4

change

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US / tʃeɪndʒ /

UK / tʃendʒ /

A1
  • noun
  • Exchange of one set of clothes for another
  • Money in the form of coins instead of paper
  • Money returned after giving too much
  • Act of making or becoming something else
  • verb
  • To exchange one set of clothes for another
  • To exchange one kind of money for another
  • To replace something with another thing
  • To make or become something else
  • To go from one train, bus, etc. and go to another

derivative

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US / dɪˈrɪvətɪv /

UK / dɪˈrɪvətɪv /

B1
  • adjecitve
  • Not from original ideas; from somewhere else

they

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US / ðeɪ /

UK / ðe /

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

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

length

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US / leŋθ /

UK / leŋθ /

A2
  • noun
  • Distance from one end of a pool to the other
  • Measurement of distance or of time

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

helpful

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US / ˈhelpfl /

UK / ˈhɛlpfəl /

A2
  • adjecitve
  • Doing things that help someone

equal

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US / 'i:kwəl /

UK / ˈikwəl /

A1
  • adjecitve
  • Same in shape, size, or number
  • verb
  • To add up or be the same as

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

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

really

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US / ˈri:əli /

UK / ˈriəˌli, ˈrili /

A1
  • adverb
  • Used to stress something that is said; definitely
  • In a true or accurate manner
  • Very

yellow

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US / ˈjeləʊ /

UK / ˈjeloʊ /

A1
  • adjecitve
  • Color of lemons or the sun
  • Lacking in courage; cowardly (used as an INSULT)
  • verb
  • To turn yellow, often due to being old

naught

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US / nɔ:t /

UK / nɔt /

B2
  • noun
  • Quantity of no importance, nonexistence

involve

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US / ɪn'vɒlv /

UK / ɪnˈvɑlv /

A2
  • verb
  • To have or be included as a part of something

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

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

give

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US / ɡɪv /

UK / ɡɪv /

A1
  • noun
  • Degree of flexibility in something, a material
  • verb
  • To hand over or present something to someone
  • To cause someone to have or experience something

against

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US / əˈgenst /

UK / əˈɡenst /

A1
  • preposition
  • In physical contact with, as for support; next to
  • In opposition to; disagreeing with

angle

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US / ˈæŋgl /

UK / ˈæŋɡəl /

B1
  • noun
  • Place where two lines or surfaces meet

same

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US / seɪm /

UK / sem /

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

relationship

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US / rɪˈleɪʃnʃɪp /

UK / rɪˈleʃənˌʃɪp /

A2
  • noun
  • Connection between two or more people or things
  • Way people interact or live with each other
  • Romantic or loving friendship between two people

color

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US / 'kʌlə(r) /

UK / ˈkʌlɚ /

A1
  • noun
  • Quality of things you can see, e.g. red, blue
  • Pink or red in your face, e.g. after being ill
  • The shade of someone's skin e.g. black, or yellow
  • verb
  • To change or affect someone's opinion
  • To make something colorful using colored pencils

not

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US / nɒt /

UK / nɑt /

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

zero

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US / ˈzɪərəʊ /

UK / ˈzɪroʊ /

A1
  • noun
  • Temperature when water turns to ice
  • number
  • Number 0
  • verb
  • To set a scale or other device to show 0

but

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US / bət /

UK / bʌt,bət /

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

definition

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US / ˌdefɪˈnɪʃn /

UK / ˌdɛfəˈnɪʃən /

A2
  • noun
  • Explanation of word's meaning, as in dictionaries
  • The sharpness of an image on a screen
  • Visual clearness in a photo or on a screen

maybe

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US / ˈmeɪbi /

UK / ˈmebi /

A1
  • adverb
  • Possibly but not certainly; perhaps

piece

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US / pi:s /

UK / pis /

A1
  • noun
  • A counter in a board game
  • Small part of something larger
  • verb
  • To put items together to assemble something

talk

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US / tɔ:k /

UK / tɔk /

A1
  • noun
  • Style of speaking
  • Discussion between two countries
  • Giving information in front of people; lecture
  • Saying things or ideas to someone with words
  • verb
  • To make a formal speech about something
  • To say things or ideas to someone with words

constant

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US / 'kɒnstənt /

UK / ˈkɑnstənt /

A2
  • adjecitve
  • Happening frequently or without pause
  • noun
  • Thing that happens always or at a regular rate

why

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US / waɪ /

UK / hwaɪ, waɪ /

A1
  • adverb
  • Question word to ask for a reason
  • exclamation
  • (Used to express surprise or approval)

per

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US / pə(r) /

UK / pɚ /

A2
  • preposition
  • In the manner directed or suggested
  • For each; during each

algebraic

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US / ˌældʒɪ'breɪɪk /

UK / ˌældʒəˈbreɪk /

C1
  • adjecitve
  • Concerning algebra

time

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US / taɪm /

UK / taɪm /

A1
  • noun
  • Speed at which music is played; tempo
  • Point as shown on a clock, e.g. 3 p.m
  • Number of hours, minutes needed to do something
  • Occasion when something happens
  • Period or occasion that something occurred
  • Period in history or the past
  • Something measured in minutes, hours, days, etc.
  • How long an event takes; duration
  • verb
  • To check speed at which music is performed
  • To choose a specific moment to do something
  • To measure how long an event takes, e.g. a race
  • To schedule something to occur at a specific moment

set

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US / set /

UK / sɛt /

A1
  • adjecitve
  • Prepared for something; ready
  • noun
  • Complete group of something
  • Place where a television show or movie is filmed
  • Group of games in a tennis match
  • Device that receives radio or television signals
  • verb
  • To make a clock state or ring at a particular time
  • To decide upon or choose something
  • To make something ready for use (e.g. table)
  • To decide on a price for something
  • To become hard (of glue, concrete etc.)
  • To be located in a specific place or time
  • To put (e.g. a vase) down carefully in a place
  • To put someone in a certain condition or state
  • (Of the sun) to go lower than the horizon

possibly

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US / ˈpɒsəbli /

UK / ˈpɑ:səbli /

A1
  • adverb
  • May be true or likely, but is uncertain

equation

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US / ɪˈkweɪʒn /

UK / ɪˈkweʒən, -ʃən /

B1
  • noun
  • Mathematical statement showing things to be equal
  • Complex set of different facts, ideas or issues

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)

for

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US / fə(r) /

UK / fɔr,fə /

A1
  • preposition
  • Used to show the purpose, or need of something
  • Subtitles section
  • A 20-meter ladder is leaning against a wall.
  • The distance x of t between the bottom of the ladder and the wall is increasing at a rate of three meters per minute.
  • At a certain instant, t sub zero, the top of the ladder is a distance y of t sub zero of 15 meters from the ground.
  • What is the rate of change of the angle theta of t between the ground and the ladder at that instant?
  • So what I'm gonna do is draw this out, and really the first step is to think about, well, what equation will be helpful for us to solve this problem?
  • And then we might just go ahead and actually solve the problem.
  • So a 20-meter ladder is leaning against a wall.
  • So let me draw ourselves a wall here.
  • That is my wall.
  • Now let me draw our 20-meter ladder.
  • So maybe it looks something like that.
  • So that is 20 meters.
  • They say the distance x of t between the bottom of the ladder and the wall, so it's this distance right over here, this is, this distance right over here is x of t.
  • They say it's increasing at a rate of three meters per minute.
  • So we know that we could either say x prime of t, which is the same thing as dx dt, is equal to three meters.
  • I'll write it out, because if I just said m per m, it might not be that clear.
  • Meters per minute.
  • So they give us that piece of information.
  • So the rate of change of x with respect to time, they give us that.
  • At a certain instant, t sub zero, the top of the ladder is a distance of 15 meters.
  • So the top of the ladder, so let's make this very clear.
  • So this distance right over here is y of t.
  • Y of t.
  • And they say at time t sub zero, y of t is 15 meters.
  • Let me just write it here.
  • Y of t sub zero is equal to 15 meters.
  • And so maybe, let me write this right over here.
  • This is y of t sub zero.
  • Let's just assume that we're drawing it at that moment, t sub zero, because I think that's going to be important.
  • Y at t sub zero is equal to 15 meters.
  • So they want to know what is the rate of change of the angle theta between the ground and the ladder?
  • And this is just saying theta is also going to change with respect to time.
  • It's going to be a function of time between the ground and the ladder at that instant.
  • So theta, let me get a new color here.
  • Theta is this angle right over here.
  • This is theta, and it's also going to be a function of time.
  • So what we'll always want to do in these related rates problems is we want to set up an equation, and really an algebraic equation, maybe a little bit of trigonometry involved, that relates the things that we care about.
  • And then we're likely to have to take the derivative of both sides of that in order to relate the related rates.
  • So let's see.
  • We want to know, we want to know the rate of change of the angle between the ground and the ladder at that instant.
  • So what we need to figure out, we want to figure out theta prime at t sub zero.
  • This is what we want to figure out.
  • Now, they've given us some interesting things.
  • They've given us, I guess, our rate of change of x with respect to time is constant at three meters per minute.
  • And we know what y is at that moment.
  • So let's see.
  • Can we create a relationship?
  • Because they gave us dx dt, it'll actually be more useful to find a relationship between x and theta and then take the derivative of both sides and then use this information possibly to figure out what the appropriate value of x or theta is at that moment.
  • So let's do that.
  • So how does x relate to theta?
  • Well, we use a little bit of trigonometry right over here.
  • If you took the hypotenuse times the cosine of theta, you would get x.
  • So let me write this right over here.
  • X of t, x of t is equal to the hypotenuse, 20 meters, that's the length of the ladder, times the cosine, cosine of theta.
  • And I could say the cosine of theta of t just to make it clear that this is a function of time.
  • This comes straight out of trigonometry, actually our basic trigonometric function definitions.
  • Now, why is this useful?
  • Why do I think this is useful?
  • Well, let's think about what happens when I take the derivative of both sides using the chain rule.
  • On the left-hand side, I am going to have an x prime of t and then that's going to be equal to, what do I end up on the right-hand side?
  • Well, using the chain rule, first I'll take the derivative with respect to theta and so that's just going to be negative 20 sine of theta of t and then I need to multiply that times theta prime of t.
  • So what I could do is say, hey, look, at t sub zero, I know what x prime of t is.
  • I could try to figure out what sine of theta of t is and then I'll just solve for this right over there.
  • So let's do that.
  • So at t sub zero, so at t is equal to t sub zero, what we're gonna have, x prime of t, well, at every time, it's three meters per minute.
  • We'll assume that our rates are in meters per minute and just our values are in meters when we're talking about distance and our angles are in radians.
  • So this is going to be equal to three is equal to negative 20 times sine of theta of t times the derivative of theta with respect to time.
  • So how do we figure out what sine of theta of t is going to be?
  • Well, let's just use that other information and I'm gonna scroll down a little bit, get a little bit more real estate.
  • So sine of theta, let me write it over here, sine of theta at time t sub naught, that's what we care about, t is equal to t sub naught, what's that going to be?
  • Well, sine is opposite over hypotenuse.
  • So that's going to be y at t sub naught over our hypotenuse of 20 meters.
  • Well, that's going to be equal to, that's going to be equal to, they tell us y of t sub naught is 15 meters over 20 meters, which is the same thing as 3 4ths.
  • So by this yellow information, they actually told us that this right over here is going to be equal to 3 4ths.
  • So this times 3 4ths times the rate of change of theta with respect to t.
  • And so now we just solve for this and we're done.
  • So this is going to be, what's negative 20 times 3 4ths?
  • That is negative 15, that is negative 15. If we divide both sides by negative 15, we get theta prime of t is equal to three over negative 15, three over negative 15, which is the same thing as being equal to negative 1 5th.
  • And the units here would be in radians per minute because our rates are all in per minute.
  • So if I wanted to, I could write radians per minute.
  • Ideally, I would write it right over here.
  • But there you go.
  • We're able to figure out, and this is an interesting one because they give you that information on y, but really use that information of y to figure out what sine of theta of t is.
  • But the equation you set up involves x and theta.