LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers

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1 LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the ppulatin f Massachusetts in the year Try an epnential regressin n the data set: 8 77 Year Ppulatin,,,, 7,8, 8,77,,,,, p 87 p 787 S, this mdel predicts the ppulatin t be apprimately 7,, nswer t questin : The ppulatin f Mass in is predicted t be abut 7,, vending machine peratr sells, cans f sda mnthly at a price f cents each She believes that fr each cent increase in price, her mnthly sales will drp by cans What shuld her selling price be if she wishes t achieve the maimum pssible revenue? Let n number f cent price increases Revenue cst per cannumber f cans sld R n n n If yu cnsider the dmain t be n, then this Revenue functin has a maimum at n, meaning that it s best t keep the price at cents If yu include als include negative values fr n in ther wrds, if yu assume that price decreases lead t a cmparable amunt f increased sales, there is a maimum f n / If it is pssible t have selling prices in between -cent increments, this wuld give an ptimal selling price f 7 cents If the price must be a multiple f cents, then cents and cents are the best pssible selling prices nswer t questin : nswers f, 7, r cents were accepted Find the cube rts f the cmple number i Yu may epress yur answer either in rectangular frm r in plar frm π In plar frm, i cis DeMivre s Therem abut rts f cmple numbers in plar frm gives the answers belw / π / πk nswer t questin : The cube rts are cis, fr k,,

2 Write parametric equatins fr the ray that meets the fllwing criteria: the ray s terminal pint is, the ray must pass thrugh the pint, t must crrespnd t the pint, e sure t state the dmain f t values Hnrs Pre-alculus Final Eam nswers page f Use, y initial pint tvelcity vectr, where the initial pint is the given,, and the velcity vectr is <, > <, > <, > This gives, y, t<, > reaking the vectr equatin int tw parametric equatins, we get: nswer t questin : parametric equatins: t y t dmain f t values: t < Given with,, and, what is the length f side? We are given SS infrmatin fr, s we can use the Law f Sines mbiguus ase sin sin sin sin sin sin sin sin S, angle is either the acute angle r the btuse angle 8 ase : If angle is, then angle is 8 Then we can use the Law f Sines t find : sin sin sin sin sin sin ase : If angle is, then angle is 8 Then we can use the Law f Sines t find : sin sin sin sin sin sin nswer t questin : r Let P ccrding t the Ratinal Zers Therem, what are the pssible ratinal zers f P? factrs f ±, ±, ±, ±, ±, ± factrs f ±, ±, ±, ± This gives the list f pssible zers shwn belw nswer t questin : The pssible ratinal zers are

3 ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ± Hnrs Pre-alculus Final Eam nswers page f

4 7 Find a mdel invlving the sine functin fr the given graph Vertically, the amplitude f and there is a shift dwnward by Mdeling just these features wuld give the frmula sin Hrizntally, the perid is, which gives a cefficient f π/ π Nw the frmula is sinπ ll that remains is t mdify the frmula t shift the graph hrizntally One pssibility is t add a shift by / unit t the right This gives the fllwing as a final mdel: Hnrs Pre-alculus Final Eam nswers page f nswer t questin 7: f sinπ / 8 Find the angle measure in radians between vectrs u and v in the given diagram [Diagram shwed vectr u frm, t, and vectr v frm, t, ] Let θ represent the angle between u and v u v <, > <, > csθ u v <, > <, > Use inverse csine t get θ radians r nther apprach: Use Law f sines n the side lengths f the triangle nther apprach: Find acute angles f tw right triangles, then add nswer t questin 8: The angle measure is radians Part Medium Prblems a Hw many ways can yu elect a -persn cmmittee frm a grup f peple? Order des nt matter, s r b Hw many ways can yu elect a president, a vice-president, a treasurer, and a secretary frm a grup f peple? The jbs must be held by different peple Order des matter, s 8 7 P c If a -persn cmmittee is chsen at randm frm a grup f peple, and if my and Katie are tw f the peple, what is the prbability that my and Katie are bth n the cmmittee? number f cmmittees cntaining bth my and Katie ttal number f cmmittees 8 r % chance Using yur calculatr, graph this equatin n the grid: y y y One way is t grup the y, y, and cnstant terms and then use the quadratic frmula t slve fr y y y y y y y y ± y y Shw the algebraic steps t find all slutins t the

5 trignmetric equatin sin sin cs Hnrs Pre-alculus Final Eam nswers page f sin sin sin w ± w sin r sin / { sin cs sin sin sin w π π π,, }, / πn nswer the fllwing questins abut the series n a Fill in the first three terms and the last three terms f the series b Is the series an arithmetic series, a gemetric series, r neither f these? There is a cmmn difference f, s it is arithmetic c Find the sum f the series, r eplain why the series des nt have a sum ll finite arithmetic series have finite sums, s n a a n Yu are given the fllwing infrmatin abut P: P is a plynmial with real cefficients The graph f P fr real numbers is shwn The nly real zers f P are and Pi Write a pssible frmula fr P Yu may leave it in either standard frm r factred frm Eplain hw yu get yur frmula Frm the given infrmatin we knw there is a zer at, and frm the graph we knw the zer at is a duble rt Since all the cefficients are real, we knw the imaginary rt's cnjugate is als a rt That gives us rts, which matches with the picture at right P i i P This a a plynmial with the apprpriate zers Our plynmial culd by scaled by any real number t stretch it vertically hecking ur plynmial with the graph f the plynmial at right, we see that it's current scaling by is reasnable 8 n

6 Hnrs Pre-alculus Final Eam nswers page f Nte: Mr Kresser s class had a different prblem fr prblem Find the partial fractin decmpsitin f Then, multiply bth sides f the equatin by t get: Equating like cefficients gives the fllwing system f equatins: We can slve this system using an augmented -by- matri: rref S,,, and Therefre, the partial fractin decmpsitin is:

7 Hnrs Pre-alculus Final Eam nswers page 7 f Part Lng Prblems prblems, 8 pints each, pints ttal nsider the given quadrilateral D a Find the missing side length and angle measures Write yur answers n the diagram First find D: D D Find <D and <D: Find D: < D sin < D 7 D Find <D: Find <D: D 7, s sin< D sin 7 < D cs 7 < D 77 b Find the area f D rea f D rea f D rea f D sin 7 8 c Let pint lie at the rigin f a artesian crdinate system with segment alng the psitive -ais Find the crdinates f a pint P, lying n diagnal D, such that rea f P rea f DP D Let DP, s that P sin < D sin < D sin sin 7 8 Draw a right triangle as shwn and use right triangle trig t find,y crdinate f P The pint is, nther pssible argument: nsider DP and P as the bases f the triangles The heights f the tw triangles t these bases are equal T have equal areas, the bases must als be equal Thus DP P, making P the midpint f D nther pssible argument: D has area, s P has area Using as the base f P, the height must be This shws P must have a y-crdinate f ; then find

8 Each part f this prblem invlves transfrmatins Hnrs Pre-alculus Final Eam nswers page 8 f a Given the graph f F n the first grid, sketch the graph f F n the secnd grid b Find the perids f the functins tan θ and tankθ Use the unit circle definitin f tangent t justify yur first answer, and use a transfrmatin argument t justify yur secnd answer The perid f tan θ is π because tan θ is defined as y/ The values f this rati fr third quadrant angles are the same as fr the crrespnding first quadrant angles, because bth and y are ppsite in the third quadrant Similarly, tan θ fr furth quadrant angles take the same values as secnd quadrant angles Therefre, the values f tan θ repeat after an interval f π The perid f tankθ is π/k because the factr f k shrinks the graph hrizntally by a factr f k c circle is described parametrically by cs t, y sin t If this circle is stretched hrizntally by a factr f, an ellipse is btained Sketch the circle and the ellipse, then write a pair f parametric equatins describing the ellipse Fr the ellipse, just duble the crdinate: cs t, y sin t

9 Hnrs Pre-alculus Final Eam nswers page f 7 Fr all parts f this prblem, suppse functins f and g are inverses f each ther a If pint, lies n the graph f g, what can be said abut the graph f f? Pint, lies n the graph f f b Re-epress fgf8 in the simplest pssible frm f8 c Given that f lg, identify g, and find g g ; g /8 d Given that f lg, re-epress fa fb in the frm f f a f b lg a lg b lg lg lg f lg a lg b a b a b a b lg lg 8 D the fllwing fr the functin f a Find the, y crdinates f any remvable discntinuities f the graph f f The ratinal functin factrs int S, there is a remvable discntinuity at The crdinates f that remvable discntinuity are:, f, b State the dmain and the range f f The dmain f f is,,, The range f f is,,, 7 7, 7 c Determine each f the fllwing limits If the limit is nt a finite number, chse ne f the fllwing answers:,, r des nt eist lim - lim lim lim lim - d Fr what values f is f increasing? f is never increasing