WYSE Academic Challenge Regional Mathematics 2007 Solution Set

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1 WYSE Academic Challenge Reginal Mathematics 007 Slutin Set 1. Crrect answer: C. ( ) ( ) 1 + y y = ( + ) + ( y y + 1 ) = ( ) ( 1 + y ) = s *1/ = 1. Crrect answer: A. The determinant is ( 1 ( 1) ) (1 1 ( 1) 1) + 1(1 1) =.. Crrect answer: C. Heagn. Draw tw radii such that anther segment f the same length cnnects their endpints. This gives an equilateral triangle. Equilateral triangles have all three angles measuring 60 degrees. Si such triangles can be fit inside the circle such that ne verte is in the center and the sides are shared with the net triangle. The 6 segments with endpints n the circle frm a heagn.. Crrect answer: B. a = 1, b = -k and c = 1 als b ac > 0 k > s k > r k < 5. Crrect answer: E. Since the rts are + i, i, + i, and i, multiply the first pair t get +1, the secnd t get + 5, and the tw pair results t get f ( ) = Crrect answer: A in. The surface area f the triangle is fund by first finding it s height. The perpendicular line drawn frm the tp verte f the triangle t the base als cuts the triangle int tw right triangles. Using the Pythagrean therem we find the height t be.6. S, the area f the triangle is ½ (.6*)=6.98 in. We then must subtract the area f the circle: π(1.5/) = π(.75) =1.767 in. S, the area f the triangle face withut the circle prtin is = in. The area f each f the rectangular side is * r 1 in. The inner area f the cylinder is the circumference times the length f the cylinder. Its length is inches. The circumference is π r = π (.75) =.71 in. The area f the inner cylinder is then.71 * = 1.17 in. The ttal surface area is then (5.161) + (1) = in. 7. Crrect answer: A. The first persn that yu ask can be brn n any mnth f the year, say the mnth f May. Then, the net persn that yu speak t must be brn in May. 8. Crrect answer: E. Certainly the direct slutin is t g t base 10, find the seventh number, then g back. The base 10 sequence is, 9, 7, 81,, 79, 187. In base eight, the seventh number wuld be 1. It is als pssible t d this using base eight multiplicatin rule 11 * = (9)(9) which after carrying ver wuld give us the crrect Reginal Slutin Set

2 9. Crrect answer: A. 10 and 0.Let the length f the side frm the first square be and the length f the side frm the secnd square be y. Then, = / y -10. We als knw that + y = Using substitutin, (/ y 10) + y = 1000 /9 y 0/ y y = /9 y 0/ y 900 = 0 Using the quadratic frmula gives y = 0. Since = / y -10, = 10. S, the sides f the squares are 10 and Crrect answer: C. E( ) = X * P( ) s $.65 = $750(1/1000) + $500(/1000) + $00(/1000) + $X(/1000) and $X = $ Crrect answer: E. Since this equatin has an y cmpnent when in the frm A + By + Cy + D + Ey + F = 0, we must lk at the discriminant B AC. The discriminant is 1, s the equatin is a hyperbla. (Incidentally, it s the hyperbla + y = 0 under a rtatin f 5 degrees.) 1. Crrect answer: D in,.5 in. First, ntice tw similar triangles: triangle ABC and triangle DEC. Segment BC has length 50 in. and segment BE has length 0 in. We can then cnclude that segment EC is 0 in. As a result f similar triangles, /0 = y/50. Slving fr gives = /5 y. The area f triangle ABC is equal t the area f trapezid ABED plus the area f triangle DEC. S, ½ (50*y) = 50 + ½(0*) 5y = Using substitutin, 5y = (/5 y) 5y = y 16y = 50 y =.5 Then, = /5 y = /5 *.5 = S the bases are 19.5 in and.5 in Crrect answer: B. Mid-pint at, =, 1 d = = 7 and d = ( ) + ( + 1) 007 Reginal Slutin Set

3 1. Crrect answer: D. If A is the length f the semi-majr ais, B is the length f the semiminr ais, and C is the fcal length, then the infrmatin given tells us AB π = 60π and C/A = 0.8. Since A = B + C, C = 0.8A, A = B + 0.6A, 0.6A = B, 0.6A = B, and A ( 0.6A)π = 60π. This means A = 10, B = 6, and C = Crrect answer: B. 768 cm. Drawing a perpendicular line frm the base t a tp verte gives a right triangle with hyptenuse 0. Yu will d this t bth sides f the trapezid. The middle segment n the lnger base minus the segments shared by these tw right triangles must be the same length as the shrter base, 1. This leaves a length f 6 cm t be shared between the legs f the tw triangles. Split the length in tw equal prtins since the triangles have the same measures. Each segment will be 18 cm. Using the Pythagrean therem gives 18 + h = 0. Slving fr h gives. The area f the triangles are.5(18*) = 16cm each. The area f the inner rectangle is *1 = 6. Cmbined, its 768 cm. 16. Crrect answer: D. ( + a) = + a + 6a + a + a ( + ) = + + 6( ) + ( ) + ( ( + ) = ) 17. Crrect answer: B. The answer is the slutin t the equatin tan θ = Crrect answer: C. 7. inches. Let the length f the bamb be y and the distance f the lwer end frm the wall be. Befre the bamb slides dwn, we have a right triangle frmed with sides 6, and y. Thus, 6 + = y. After the bamb slides dwn, there is a new right triangle frmed with sides 0, + 10 and y. Thus 0 + ( + 10) = y. Using substitutin fr y, 6 + = 0 + ( + 10) = = 0 = 8.8 Then, t find the length f the bamb (y), slve: 6 + = y. y = = = 75. and y = Crrect answer: C. y = *tan(8. ) = Reginal Slutin Set

4 0. Crrect answer: B. The equatin prduces a circle with radius, which wuld have an area f 9 π. T prve this, we can cnvert the equatin int rectangular crdinates as fllws: r = 6csθ r = 6r csθ + y = y = 9 ( ) + y =. 1. Crrect answer: B. 5. Since m B= 65, m AED = 65. Since m EDH = 110, m EDG= 70 since tgether they frm a straight line. Since the sum f the angles f a triangle is 180 degrees, m G = 5. Then, m C = 5 since it is the crrespnding angle t angle G in the cngruent triangle. Crrect answer: A. If the first child gt A, the secnd B, the third C, and each invested, then we end up with A + B + C = 197,00, 1/5A =, 1/B =, and 1/C =. After substituting and slving, we get = 197,00, s = The ttal invested wuld be, r 9,50.. Crrect answer: A. y 5 1/ y z ( 1/ z / yz ) 11/ / 6 = y z y = 1/ / y z y 1/ 1/ z z / 5 / 5 / 6 1/ 6 1/ y z = yz. Crrect answer: C. Each minute, Susan is srting 1/0 th f the pallet, and Tammy is srting 1/5 th f the pallet. Susan already had 5/0 f the pallet dne. The remaining 5/0 will require t minutes, where (1/0 + 1/5)t = 5/0, s t = 18.5 minutes. 5. Crrect answer: D. 0 apples. Let represent the amunt f apples Billy picked. He gave 1/ t his first friend. He then had -1/ r / apples left. He gave ¼ f the remaining apples t his secnd friend. S, he then had / 1/*/ = /6 1/6 = /6 = 1/ apples left. T his net friend he gave 1/5 f his remaining apples. This left 1/-1/5*1/ = 1/-1/10 = 5/10 1/10 = /10 = /5 apples left. Since he gave away apples, /5 =. This gives us = 0. S, he started with 0 apples. 6. Crrect answer: B. 15(1 + i *.5 ) = 1000 (1 + i 1 ) = 8 1 lg(1 + i ) = lg(8) lg( 1+ i ) = i = i =.6 7. Crrect answer: D. This is an eercise in cunting techniques. There are 6 equally likely utcmes. 0 have a result f three heads, 15 have fur, 6 have five, and 1 has si. The verall prbability is therefre /6 = Nte that this can als be slved using binmial prbabilities. 8. Crrect answer: E. Let represent the height f the stump remaining in the grund. Then the prtin f the tree that was brken ff was 18. This gives us a right triangle with legs and 6 with hyptenuse 18. Thus: + 6 = (18 - ), + 6 = = 88 = 8. S, the tp segment f the tree was 18 8 r 10 feet lng. 9. Crrect answer: E. (1/)(0/7) + (1/)(0/) + (1/)(1/) = 1/9 007 Reginal Slutin Set

5 0. Crrect answer: D. First we need the time the ball hits the grund. This wuld be the slutin t t + 0 = 0, which is t = The average velcity is an average rate f h(1.18) h(0) change frm t = 0 until t = 1.118, which wuld be = Crrect answer: B. 5. Angles C and D cut ff the same arc. S, + = 5-0. Slving this gives = 1. Placing this back int ne f the epressins, such as 5-0, gives us the measure f the angle: 5(1)-0 = 0. Since the sum f the angles f a triangle are 180, m B = 180 and m B= Crrect answer: C. = ( 5), s Limit 5 = 7 +. Crrect answer: E. = ( + 1) ( + 1)( 1) ( 1) = + ( + 1)( 1) ( + 1)( 1) ( + 1) = + ( 1) + = + = 0 ( + 1)( ) = 0 = 1,, but -1 is an etraneus slutin.. Crrect answer: E. = 1 Squaring bth sides gives: ( ) + ( )( + 1) + ( + 1) = ( )( + 1)) = 5 + ( )( + 1)) = 1 + Square bth sides again: ( )( + 1) = = = 0 Factr int ( - 1)( + 1) = 0 t yield answers f = 1 r = -1. Hwever, if = -1, there will be a negative under the radical. S, the slutin is Crrect answer: A. Bb makes a yellw ctagn (II matches by t shape, and ctagn can t be green by II, blue by I, r red by III). At this pint, Den s shape must be red (red can t be Charlie s by II r Aarn s by III), s Aarn s shape is a blue square (blue by eliminatin, square by I). This must mean that Den s shape is a red triangle (less sides than square by III). By default, Charlie s shape is a green heagn Crrect answer: C.. 10 = = 10 s ph = Reginal Slutin Set

6 7. Crrect answer: C. The functin is f the frm Acs(B), s the perid equals π π = = π. B 8. Crrect answer: A. 00 Multiplying gives y = Cmplete the square: y = ( ) = ( 00) y = ( 00) y = ( 00) The maimum yield is apples when there are 00 trees. Mathematics 9. Crrect answer: D. area = s( s a)( s b)( s c) where s = ( a + b + c) / 18 (18 1)(18 8)(18 16) = 160 = Crrect answer: C. 1 lg = 1+ lg lg = lg10 + lg lg = lg = = = 0 ( + )( 1) = 0 Which gives us =- and =1. But, lg(-) is nt defined, s the nly slutin is Reginal Slutin Set

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