0 k?j'..e L-+, v\.l.s t) So ~ a,pp ij ct-vh CJYIJ?t-J --1 h8-' ~ ~ erv-i

Size: px

Start display at page:

Download "0 k?j'..e L-+, v\.l.s t) So ~ a,pp ij ct-vh CJYIJ?t-J --1 h8-' ~ ~ erv-i"

Jonas Cobb
5 years ago
Views:

1 fv1 ~/1ti 1 D 7, f; V,~vi t=l fi &'-Y) 0 k?j'..e L-+, v\.l.s t) So ~ a,pp ij ct-vh CJYJ?t-J --1 h8-' ~ ~ erv-i o chred

2 Math Variation Objectives: 1) Direct variation 2) nverse variation 3) Joint variation Key words for recognizing variation problems: "varies" or "proportional" Direct Model Words Translation to Model equation Graph of model if math k=2 "y varies directly :1; as x" ''y varies" is always y=k "y is directly y=kx proportional to x" "directly" means x is in the numerator ~ 1.' "y varies as x" ofrhs ' ' '. "y varies" is always "y varies inversely y=k 1 as x" y=k - x \~_ ~~. ~ nverse "inversely" means or ' ' "y is inversely xis in the k proportional to x" y=denominator of -1~' x RHS "Joint" means that "y varies" is always 3 or more y=k variables are involved. "directly" means x and w 2 are in the X W 2 Not a 2-dimensional Joint Example: "y numerator of RHS y=k -- varies jointly as x z Vv graph and the square of "inversely" means w, and inversely z and Vw are in as z and the cube root ofv" the denominator of RHS Cautions: Direct variation problems can be solved by proportions. nverse and Joint variation cannot! Variation equations have only multiply and divide, never add or subtract. Always write units on the final answer. Use units to help identify which values go with which variables. Process that works for ANY type of variation problem: Step 1: Define any variables, if needed. {Use letters that make sense.) Step 2: Translate the sentence into an equation of variation. Don't forget kl Step 3: Substitute a complete set of numbers (given in question) and solve fork. Step 4: Solve the incomplete set of numbers (given in question) and solve to answer the question.

3 Examples ~ 1. Hooke's law states that the distance a spring stretches is directly proportional to the weight attached to the spring. f a 40-pound weight attached to a spring stretches the spring 5 inches, find the distance that a 65 pound weight will stretch that same spring. 2. Boyle's law says that if the temperature stays the same, the pressure P of a gas is inversely proportional to the volume V. f a cylinder in a steam engine has a pressure of 960 kilopascals when the volume is 1.4 cubic meters, find the pressure when the volume increases to 2.5 cubic meters. 3. The lateral surface area of a cylinder varies jointly as its radius and height. a. Express this surface area Sin terms of radius rand height h. b. f the lateral surface area is 20Jr square cm when the radius is 2 cm and the height is 5 cm, find the exact constant of variation and the equation of variation. c. Find the radius when the lateral surface area is 40Jr square cm and the height is 2 cm. 4. The maximum weight that a circular column can support is directly proportional to the fourth power of its diameter and is inversely proportional to the square of its height. A 2-meter-diameter column that is 8 meters in height can support 1 ton. Find the weight that a!-meter-diameter column that is 4 meters in height can support.

4 Extras: 1. The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as its length. f a beam Vi foot wide, 1/3 foot high, and 10 feet long can support 12 tons, find how much a similar beam can support if the beam is 2/3 foot wide, Vi foot high, and 16 feet long. 2. The horsepower to drive a boat varies directly as the cube of the speed of the boat. f the speed of the boat is to double, determine the corresponding increase in horsepower required. 3. The volume of a cone varies jointly as its height and the square of its radius. f the volume of a cone is 32.7l' cubic inches when the radius is 4 inches and the height is 6 inches, find the volume of a cone when the radius is 3 inches and the height is 5 inches. 4. The intensity of light (in foot-candles) varies inversely as the square of x, the distance in feet from the light source. The intensity of light 2 feet from the source is 80 foot-candles. How far away is the source if the intensity of light is 5 foot-candles?

5 "~ ~! 1'\~o V~!o...+\o;'- "--~ frob\~ So\.., ivu} \) 'l)\ rec...\- 'ilt.j\~+ 1 OV\ ~""'\ '"'~e., ~~0-1\1 ~.L \ \~ f\ ~ - vo -AOT ~o \rt... vo'j 3; 0 o, vd'- yr:u-\cxn CTY"l k~ U:><>r<l: ~ r&"fl"''v ""} -\- c~ \"'"'1o1~~ ~ '* v ()Jr \-e.5.,_ y~urho~\. \-lo ww~, So~ \AS\~ " f "'ro<--l-i ~ ~.s-k ~ -1"4=- f..-o lo\~ sl..o pi-orur..hov\~j \.ov.-\ ~ ~e..r-\<:s~ ~e.. C..OvV\ \:,.a.. 5o~l, D rd,{ ~""';j-- 'f''""f <>' -l.gy's. G-.oo..t '. o"'-e-- VV\.t.~o~ ~ oil ~ ~~.e.c; 0~ \NU'"'\~ fra 'f100.\-- sk.e, Trt"_.v,-::o\o-3-e... ~~\Ve- P~ '\J'O.Jr\j~ \b OJ'\~~ "1)\!"ec\-, f ~\oj-\&./\ '{Y>.oJ.e.\ ~vo\o\~ '. e-1" ' VCA-v-\~s d\<e.c..4- \' ~~ X ' \s J1ru~\j <Pvof>o-c~ crncj --Th x ~ \".> ~ V'-~N\\oe," > CA... C...o"'-s~k- / ~\.\.e..o\ P>s\\ 'JOU-\aXAo~ r-ro \o\~s "' '"V-e. \<_. ~ \ 'S -e-~:/;,e_,~-\-\~ \. ~e.. n -ro' \_,, -~"',,,,, ' 1 \o \ _.,, \ \, 1'...,_.,\ \" O.._. ~"'--t-.e_ o\.o \/\ 4..- V.J -\-\A 0 \..L-~ "1"!"", Dv'\. '"1: >r\ '\/'e'<':>a- \/ ()...t- \ ~-\-\ 0 V\ Mo~e.\?v--o'k>\eM" ' ~\.es \V'\~~~ "'-s?< w. y \ s )V\~rs.t.-~ <pv-o~e<hoy"\j 4o?<,, ;! ' />is tr<- ~ ""6::;. \<:. j_ /~ ~ \ V\ ~ev\q rl'\.l ~ ~ o~ f<.tts ff'r ~:;.- h ~ \ 11\.CV' e c...s.e. s ' ~: 5~~\~ y-tyjr\ ~~ \/V\.e_{A_)r\S t e ~ &Y W\O '('e_.. VCJ..)'-\ A.)o \-e.,~ \ ~ V\o.\-- ra. "r>.>./1 ak, \e,.} B~ v(}ylcjo\~ \s e-\-\-\a.~": ( \"' ~OV\A\.t"\.~l 0 -:t.+?ro'o\t.nv' Jo.es V\Q+ S ':'J "'"~,,~" e--r"'lv'\~se.'~ ~ r.e-c)o; ra-vv>\ \N..,. ~ ~.L \~ \"' ~ -e_ ""V1.-Wl-f """-+u-r. ',..,...s \

6 e::.><.~\..es ot So\""~ ~\ -~o"' e'\u~*'oy'\"'o>. ~ 0 ' '\f()jr\ ~s '(ro\\a~ ~ ~ ~ 1= ""'~~ \ "/ = \<-.. x. "'i:-j ~ roc.e':>s ~ So\"\~ "1'o.Jr-\ct..--\-\ 9-t\ ~f'o lo\~ ~f \ : --\ ""-"'"' \,,._,\ e. -\"o "'-"',_e't"' o.\._c,,,, o~ " '" \ ""' ' o "', ""\ ~ \<.. ' ~';)..: So\..e. ~ \.<..,,.., ~ 1-1"' -\'A e. 'i' oro lo\..,.,,.. 'j o v.. ""' \\ \o <- ~ '"""" ' c.o-vv. f> \ e. rt se, 1.ft- V\ V--~ \oe,<'7 o'v"\e ~ ~c,~ V'd-'r- \ \ e,!?\""} dvv..,,._\,\ i A > o,_.,,,\ <;,. p \ V.Q.. ~ <>..-es..u-1- --(',,.,- \<_, i,.)<' V"c..e. w., 'Kvi.o..v ~... ~e. ~- k., we-~ Lt.Se i~ ~ ~\\ 0 ~":> : A "'~v,.>e,r ~~ "b""-"';ti.,.,,.,. ~ ~e --re.~.\- o~ ~~ -pco\o\e.vv'\. 'F. ~ ~ y-,.-o lo \-e.""'> ~ o u....,_, \ l \ bl <} v-e "' "'- peer~.,._\ S-e-. *1 1 1+! 0. V'u..lM 'o 2.-K, one. +,,-c -e. ~ ~.,0., \.<. -e.~~1,- ~ "--'\r-* "'" '!'\~ ~e.w\.,_\,\ \A ~\"--"& \v. ~ ~ o~ ',<. ~o-iv'.16\ef '2, ~?o \N-. ~ -V\A... e.. '<' v._.e...& \-e_j Vou-1 ct--6 \e, 1!! j ~~""' '(\ ~$ ~ ~<>Jr\~~ ~O"Vl--3 ~~ ~ \N\w.\-ip~ 1 V'«"'.e... \S ~ ~ av,\.lb~~,r i i

7 ...,, -~.. ~~..--~ Math 70 Practice Problems for Variation 1. Hooke's law states that the distance a spring stretches is directly proportional to the weight a ~ spring. f a 40;ound weight attached to a spring stretches the spring 5 inches, filf d the distan pound weight will stretch that same spring. l _?kp \ : D== \< vj l:)~ d ism,~ VJ~ w.e,\~k+ skp'2-: ~?>'. ~ k. 41> t=~ -:be= ~ b5 ~ l~ 4~fi. \~~ J i ii ~tothe itta65 i 2. Boyle's law says that if the temperature stays the same, the pressure P of a gas is inversely p - the volume V. f a cylinder in a steam engine has a pressure of 960 kilopascals when the vof cubic meters, find the pressure when the volume increases to 2.5 cubic meters. ~l: 'P=-~...L y ~e?. et fod ::::.!:_ l-~ "1.DLflf ::: ~ /,,.---..,,' 3. The lateral surface area of a cylinder varies jointly as its radius and height. a Express this surface area S in terms of radius r and height h. b. f the lateral surface area is 20tr square cm when the radius is 2 cm and the height i. Cl!ll, find the exact constant of variation and the equation of variation. c. Find the radius when the lateral surface area is 40tr square cm and the height is 2 o.~~l: fs::::-\<. r k l. ' -;:. eez : dolf:::- ~ :;;l's 1~11=~ 'l..ew..-1-\e ~"' ( S =- 2.1Tft...h J c.. ::::.--.sk:.e?> : 4-o1f =-?.-11 k. ~ l l0 'Y' = /l- { - : 4. The maximum weight that a circular column can support is directly proportional to the fo~ power of its diameter and is inversely proportional to the Square of its height. A 2-meter-diameter colwttt that is 8 meters in height can support 1 ton. Find the weight that a -meter-diameter column that is f iln.eters in height can support. w fl\ " ::::;;. w rf.il().~ W.ttAAA"- t ~oovie-r- d.\a.rm-ea-e..r diet~ :::-c\ (.,.o { ~ VV\ " Oo'\ ru.-r\':1 f "', ' tt; Sv.-~ 'r"of ri \ ovuj l.ave,,~ll'f S''\~ ol ~e,\~'-'ii- \,,eifl~t:::::-~ /\ cl-f"_,.,\: w ;:::- )o: JL\ : ~ \,.,~ 1'1 ; i ~.2. : cl::.-;).._ 1 _:--... \ :::- \<..t- _... ~ \ :::- \(o \<::- -. Lu -::: \{p\<.. -::9 'f::_- 4 ii< j - '- -::. '6,...,, '-"' :! V\ '8' 2. lot.\ 1: i \A)$-\ iii!;,i

8 ~ilktj!;. j '

6.7 Variation and Problem Solving. OBJECTIVES 1 Solve Problems Involving Direct Variation. 2 Solve Problems Involving Inverse Variation.

6.7 Variation and Problem Solving. OBJECTIVES 1 Solve Problems Involving Direct Variation. 2 Solve Problems Involving Inverse Variation. 390 CHAPTER 6 Rational Epressions 66. A doctor recorded a body-mass inde of 7 on a patient s chart. Later, a nurse notices that the doctor recorded the patient s weight as 0 pounds but neglected to record