Log1 Contest Round 1 Theta Equations & Inequalities. 4 points each. 5 points each. 7, a c d. 9, find the value of the product abcd.
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1 Log1 Cotest Roud 1 Theta Equatios & Iequalities Name: poits each 1 Solve for x : x 3 38 Fid the greatest itegral value of x satisfyig the iequality x x xy71 Fid the ordered pair solutio to the system of equatios x7 y109 1 How may real solutios does the equatio l x have? x Fid the sum of the solutios of the equatio 3 x 0x 100x 1 0 Solve for x : x x log 3 log 1 poits each 7 Give the equatios a bc, a bd 7, ac d 8, ad bc d 9, fid the value of the product abcd 8 y8x17 Fid the area of the regio defied by the system yx3 x y9 9 Withi 0 years, i what year is the equal sig (=) geerally attributed as beig first used i prit to represet equality? 10 Give the equatios x y z, x 3y z 8, ad 7x y z 3, fid the value of 1x 13 y z poits each 11 Fid the sum of the squares of the solutios of the equatio 3 x 9x x x x If si xcos x 1, fid the value of six 13 A right rectagular prism measures x y z, with all legths measured i cm If the space diagoal of the prism has legth cm, ad if x y z cm, fid the total surface area of the prism, i cm 1 A ellipse has mior axis of legth, ad the eccetricity of the ellipse is umerically equal to the legth of the ellipse s latus rectum Fid the ellipse s major axis legth 1 Fid the oblique asymptote of the graph with equatio writte i slope-itercept form y 3 x 3x 17x x 3 x 3x x 1,
2 Log1 Cotest Roud 1 Alpha Equatios & Iequalities Name: 1 Solve for x : x 3 38 poits each Fid the greatest itegral value of x satisfyig the iequality x x xy71 Fid the ordered pair solutio to the system of equatios x7 y109 3 Fid the sum of the solutios of the equatio x 0x 100x 1 0 Solve for x : x x log 3 log 1 poits each Give the equatios a bc, a bd 7, ac d 8, ad bc d 9, fid the value of the product abcd 7 8 The ellipse with equatio 7x 3xy 13 y 1 is geerated by rotatig couterclockwise a ellipse with horizotal major axis Fid the least positive radia measure agle by which this could be achieved y8x17 Fid the area of the regio defied by the system yx3 x y9 9 Withi 0 years, i what year is the equal sig (=) geerally attributed as beig first used i prit to represet equality? 10 Fid the sum of the squares of the solutios of the equatio 3 x 9x x x x If si xcos x 1, fid the value of six poits each 1 A right rectagular prism measures x y z, with all legths measured i cm If the space diagoal of the prism has legth cm, ad if x y z cm, fid the total surface area of the prism, i cm 13 How may ordered quadruples of o-egative itegers x, y, z, w satisfy the equatio x y z w 1? 1 A ellipse has mior axis of legth, ad the eccetricity of the ellipse is umerically equal to the legth of the ellipse s latus rectum Fid the ellipse s major axis legth 1 Fid the oblique asymptote of the graph with equatio writte i slope-itercept form y 3 x 3x 17x x 3 x 3x x 1,
3 Log1 Cotest Roud 1 Mu Equatios & Iequalities Name: 1 Solve for x : x 3 38 poits each Fid the greatest itegral value of x satisfyig the iequality x x Fid the sum of the solutios of the equatio 3 x 0x 100x 1 0 Fid the umerical value of dy dx x, y 3, whe x xy y 11 The ellipse with equatio 7x 3xy 13 y 1 is geerated by rotatig couterclockwise a ellipse with horizotal major axis Fid the least positive radia measure agle by which this could be achieved 7 poits each y8x17 Fid the area of the regio defied by the system yx3 x y9 i lim si 1 i1 is the area eclosed by y si x, the x-axis, ad what two vertical lies? 8 Let A, B, C, ad D be the solutios of A B C D Fid the value of 3 x x x x B C A D , where 9 Withi 0 years, i what year is the equal sig (=) geerally attributed as beig first used i prit to represet equality? 10 Fid the sum of the squares of the solutios of the equatio 3 x 9x x x x 9 0
4 poits each 11 What is the greatest real value of k such that si x cos x k has a real solutio? 1 A right rectagular prism measures x y z, with all legths measured i cm If the space diagoal of the prism has legth cm, ad if x y z cm, fid the total surface area of the prism, i cm 13 How may ordered quadruples of o-egative itegers x, y, z, w satisfy the equatio x y z w 1? 1 A ellipse has mior axis of legth, ad the eccetricity of the ellipse is umerically equal to the legth of the ellipse s latus rectum Fid the ellipse s major axis legth 1 dy Fid the solutio to the equatio xy form y f x dx through the poit,e, writte i the
5 Log1 Cotest Roud 1 Theta Equatios & Iequalities Name: poits each 1 Solve for x : x Fid the greatest itegral value of x satisfyig the iequality x x 3 xy71 Fid the ordered pair solutio to the system of equatios x7 y109 1 How may real solutios does the equatio l x have? x Fid the sum of the solutios of the equatio x 0x 100x 1 0, 1 Solve for x : x x poits each log 3 log Give the equatios a bc, a bd 7, ac d 8, ad bc d 9, fid the value of the product abcd 8 y8x17 Fid the area of the regio defied by the system yx3 x y9 9 Withi 0 years, i what year is the equal sig (=) geerally attributed as beig first used i prit to represet equality? 10 Give the equatios x y z, x 3y z 8, ad 7x y z 3, fid the value of 1x 13 y z 1 17 (accept ay year from 137 to 177) 0
6 poits each 11 Fid the sum of the squares of the solutios of the equatio 3 x 9x x x x If si xcos x 1, fid the value of six 13 A right rectagular prism measures x y z, with all legths measured i cm If the space diagoal of the prism has legth cm, ad if x y z cm, fid the total surface area of the prism, i cm 1 A ellipse has mior axis of legth, ad the eccetricity of the ellipse is umerically equal to the legth of the ellipse s latus rectum Fid the ellipse s major axis legth 1 Fid the oblique asymptote of the graph with equatio writte i slope-itercept form y 3 x 3x 17x x 3 x 3x x 1, yx
7 Log1 Cotest Roud 1 Alpha Equatios & Iequalities Name: poits each 1 Solve for x : x Fid the greatest itegral value of x satisfyig the iequality x x 3 xy71 Fid the ordered pair solutio to the system of equatios x7 y Fid the sum of the solutios of the equatio x 0x 100x 1 0, Solve for x : x x log 3 log poits each Give the equatios a bc, a bd 7, ac d 8, ad bc d 9, fid the value of the product abcd 7 8 The ellipse with equatio 7x 3xy 13 y 1 is geerated by rotatig couterclockwise a ellipse with horizotal major axis Fid the least positive radia measure agle by which this could be achieved y8x17 Fid the area of the regio defied by the system yx3 x y9 9 Withi 0 years, i what year is the equal sig (=) geerally attributed as beig first used i prit to represet equality? 10 Fid the sum of the squares of the solutios of the equatio 3 x 9x x x x (accept ay year from 137 to 177) 9
8 11 poits each If si xcos x 1, fid the value of six 1 A right rectagular prism measures x y z, with all legths measured i cm If the space diagoal of the prism has legth cm, ad if x y z cm, fid the total surface area of the prism, i cm 13 How may ordered quadruples of o-egative itegers x, y, z, w satisfy the equatio x y z w 1? 1 A ellipse has mior axis of legth, ad the eccetricity of the ellipse is umerically equal to the legth of the ellipse s latus rectum Fid the ellipse s major axis legth 1 Fid the oblique asymptote of the graph with equatio writte i slope-itercept form y 3 x 3x 17x x 3 x 3x x 1, yx
9 Log1 Cotest Roud 1 Mu Equatios & Iequalities Name: poits each 1 Solve for x : x Fid the greatest itegral value of x satisfyig the iequality x x Fid the sum of the solutios of the equatio 3 x 0x 100x 1 0 Fid the umerical value of dy dx x, y 3, whe x xy y 11 The ellipse with equatio 7x 3xy 13 y 1 is geerated by rotatig couterclockwise a ellipse with horizotal major axis Fid the least positive radia measure agle by which this could be achieved 7 poits each y8x17 Fid the area of the regio defied by the system yx3 x y9 i lim si 1 i1 is the area eclosed by y si x, the x-axis, ad what two vertical lies? 8 Let A, B, C, ad D be the solutios of A B C D Fid the value of 3 x x x x B C A D , where 9 Withi 0 years, i what year is the equal sig (=) geerally attributed as beig first used i prit to represet equality? 10 Fid the sum of the squares of the solutios of the equatio 3 x 9x x x x x1, x (accept ay year from 137 to 177) 9
10 poits each 11 What is the greatest real value of k such that si x cos x k has a real solutio? 1 A right rectagular prism measures x y z, with all legths measured i cm If the space diagoal of the prism has legth cm, ad if x y z cm, fid the total surface area of the prism, i cm 13 How may ordered quadruples of o-egative itegers x, y, z, w satisfy the equatio x y z w 1? 1 A ellipse has mior axis of legth, ad the eccetricity of the ellipse is umerically equal to the legth of the ellipse s latus rectum Fid the ellipse s major axis legth 1 dy Fid the solutio to the equatio xy form y f x dx through the poit,e, writte i the y e x
11 Log1 Cotest Roud 1 Equatios & Iequalities Solutios Mu Al Th Solutio x 338 x 3 x 7 3 3x 7 x 1 3x x 70 3x x 3x x x 3 Therefore, the greatest itegral value of x satisfyig the iequality is Subtractig the first equatio from the secod yields x y 38 x y 7 Subtractig this secod equatio from the first origial equatio yields y Pluggig this i to ay equatio yields x, so the solutio is, 3 1 Examiig the graphs of y l x ad y, the first exists oly i quadrats I ad IV x while the secod exists oly i quadrats I ad III, so quadrat I is where a itersectio poit could exist Secodly, the first graph icreases without boud while the secod graph decreases asymptotically toward the x-axis, so there ca oly be oe itersectio poit 0 The sum of the solutios is Differetiatig implicitly, dy dy x x y 0 dx dx dy dy dy yields 3 0 dx dx dx, ad pluggig i the poit 3, x, y3, x, y3, x, y3, x 3 x 3 log 1 x 31x 1 0 x 1x 13 Usig the x1 x1 quadratic formula, x 8 77 Sice both values are greater tha 1, both logarithms are defied, ad thus both are solutios 7 Summig the four equatios yields 3 a bc d 30 a bc d 10 Subtractig each of the give equatios from this last equatio yields d, c 3, b, ad a 1 Therefore, the product is abcd ta The regio is a triagular regio with vertices at the poits, 1, 3,7, ad, Usig the shoelace method, the eclosed area is 1 A
12 7 b ba b a i Sice f xdx lim f a i a i1, lim si 1 i1 represets the area eclosed by y si x (sice f 0 always), the x-axis, x a, ad x b, where a 1 ad ba b 3 Therefore, the two vertical lies are x 1 ad x x x 3 x 1x 80 x x x x, so A, B, C, ad D Therefore, BC A D Robert Recorde is geerally attributed with usig the symbol = to represet equals i his book The Whetstoe of Witte i If you take the first equatio, subtract the secod equatio from that, the add twice the third equatio, the result is 1x 13 y z 0 Alterately, the solutio of the system is,7,, ad 1x 13 y z Usig the Newto s sums method, the sum of the solutios is S1 9, ad the 1 sum of the squares of the solutios S satisfies S 9S1 0 S six si x cos cos x si si x cos x 1 11 This is the same as askig what the maximum value is of the fuctio f x x x f ' x cos x si x f ' x 0 if cos x si x Therefore, the si cos ad maximum occurs whe cos x ad si x Sice the diagoal has legth x y z,, makig the maximum value of f 8 x y z Therefore, 7 8 x y z x y z xy xz yz xy xz yz xy xz yz 19, ad xy xz yz is the surface area 11 1 The total umber of o-egative itegral quadruples is Let aad b be the semi-major ad semi-mior axis legths, respectively, ad let c be the distace from either focus to the ceter of the ellipse Sice the latus rectum legth b c equals the eccetricity, b c Additioally, sice this is a ellipse, a a a b c b b Therefore, a b b a b b b b 1 Therefore, a 1 17, makig the major axis legth x x87 x 3x 17x x 9 x, so the oblique asymptote is 3 3 x 3x x 1 x 3x x 1 9 yx
13 1 dy x xdx l y x c y Ae y Sice the graph passes through the poit,e, e Ae Ae A Therefore, the solutio is y e x
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