2007 Academic Challenge

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7 cademic Challenge MTHEMTICS TEST - STTE FINLS This Test Consists of 4 Questions Mathematics Test Production Team Tom Ferry, Illinois State University uthor/team Coordinator Kevin Boyer, Illinois

1 7 cademic Challenge MTHEMTICS TEST - STTE FINLS This Test Consists of 4 Questions Mathematics Test Production Team Tom Ferry, Illinois State University uthor/team Coordinator Kevin Boyer, Illinois State University uthor nne Schmidt, Illinois State University uthor Sandra Co, Kaskaskia College Reviewer Don Wayman, WYSE Coordinator of Test Production GENERL DIRECTIONS Please read the following instructions carefully. This is a timed test; any instructions from the test supervisor should be followed promptly. The test supervisor will give instructions for filling in any necessary information on the answer sheet. Be sure you understand and correctly follow these instructions. You are to indicate your answer to each question by marking an oval that corresponds to the correct answer for that question. Only one oval should be marked to answer each question. Multiple ovals will automatically be graded as an incorrect answer. Be sure ovals are marked as. Not as,,, etc. If you wish to change an answer, erase your first mark completely before marking your new choice. You are advised to use your time effectively and to work as rapidly as you can without losing accuracy. Do not waste your time on questions that seem too difficult for you. Go on to the other questions, and then come back to the difficult ones later if time remains. *** TIME: 4 MINUTES *** DO NOT OPEN TEST BOOKLET UNTIL YOU RE TOLD TO DO SO! 7 Worldwide Youth in Science and Engineering WYSE, Worldwide Youth in Science and Engineering and the WYSE Design are service marks of and this work is Copyright 7 Board of Trustees of the University of Illinois at Urbana - Champaign. ll rights reserved.

2 WYSE cademic Challenge Mathematics Test (State Finals) 7. Given the following matri (where ): What would be the inverse matri? a) b).5 c) d) e) Joe saves $5/month for 45 years. t that time he begins to withdraw 4 equal monthly payments and after the 4 th payment, his account will be zero. ssuming 6% compounded monthly about what will his monthly retirement be? (Round to the nearest dollar.) a) $6,8 b) $4,63 c) $5,7 d) $4,899 e) $4, What result do we get if we take the inverse matri and multiply it by an identity matri I? a) the original matri b) the inverse matri c) the transpose matri T d) the identity matri I e) impossible to tell 7 State Finals

3 4. pentagon is inscribed at the midpoints of a pentagon, which is inscribed at the midpoints of a third pentagon as shown. Mathematics Determine the ratio of the perimeter of the outermost pentagon to the perimeter of the innermost pentagon. Please round the first number of your ratio to the nearest tenth and make the second number. a). to b).3 to c).5 to d).7 to e). to 5. For what values of c does the equation ( c) have solution(s)? a) c b) c c) c d) c e) c R π π 6. Find the following product for α, n + sin α + sin α + sin α sin α +... ( ) 4 6 n+ n ( tan α + tan α tan α ( ) tan α +...) a) b) cos c) cot d) sec α α α e) cosec α 7 State Finals

4 Mathematics 3 7. Determine the surface area of a regular octahedron that has a volume of cubic feet. regular octahedron is a three dimensional solid with eight equilateral triangles for faces. n eample is shown below. (Please round your answer to the nearest square foot.) a) sq feet b) 4 sq feet c) 346 sq feet d) 447 sq feet e) 57 sq feet 8. n old vase has a circular bottom of diameter of 8 inches. For 5 inches above this, the diameter epands evenly to a middle diameter of inches. For the net 4 inches, it tapers evenly down to a circle with diameter of 4 inches. Finally, the vase etends up inches as a cylinder. (The picture below shows a cross section of the vase.) Find the volume of this vase. (Round your answer to the nearest cubic inch.) 4 in in 4 in in 5 in 8 in a) 55 cu inches b) 44 cu inches c) 89 cu inches d) 4 cu inches e) 567 cu inches 9. Eight equally sized shaded circles are inscribed into a rectangle as shown below. Determine what percentage of the area of the rectangle is shaded (i.e., taken up by the circles). (Round your answer to the nearest percent.) a) 67% b) 7% c) 74% d) 77% e) 8% 7 State Finals

5 Mathematics 4. boy has feet of fence. What is the maimum area he can enclose using this amount of fencing? (Round your answer to the nearest whole square foot.) a) 65 square feet b) 7 square feet c) 796 square feet d) 59 square feet e) 5 square feet. Find a, knowing that matri aa B a has no inverse for any value of. a) 3 b) c) d) - e) - and. In the following diagram, points and C are each the center of their respective circles. The radius of circle ( B ) is twice as long as the radius of circle C (CB ). D and E are each on lines that are tangent to circles and C respectively. G D F E B C Determine the ratio of the length of DG to the length of GE. (Make the second number in the ratio. Round the first number in the ratio to the nearest tenth.) a).5 to b).6 to c).7 to d).8 to e).9 to 3. Three of the interior angles of trapezoid BCD have values -9, 4+, and 6+3 (angles are not listed in any particular order). Determine all possible solutions for. a) 4.5, 9.5 b) 6.5, 3.5 c) 9.5, 3.5, 3.5 d) 4.5 < < 3.5 e) 4.5 < < State Finals

6 Mathematics 5 4. Identify the following conic section: r sin θ + sinθ + r a) Ellipse b) Hyperbola c) Parabola d) Circle e) Two crossing lines 5. grain bin is shaped like a downward facing circular cone as shown below. The diameter of the cone is 3 feet and the depth is feet. Grain is pouring out of the bottom of the cone at a rate of 5 cubic feet per minute, which is causing the depth of the grain inside to fall. t the moment when the depth of the gain is 5 feet, how fast is the depth of grain falling? (Round your answer to the nearest hundredth of a foot per minute.) 3 feet feet a). feet per minute b). feet per minute c).3 feet per minute d).4 feet per minute e).5 feet per minute 6. regular octagon has a perimeter of feet. Determine its area. (Round your answer to the nearest square foot.) a) 9, square feet b) 8,64 square feet c) 5,3 square feet d) 3,4 square feet e) 57,5 square feet t 7. Given that f( ) e dt, find a) b) e + e 4 4 c) e e 4 d) 8e 4 e) 4e 4 4 f ( ). 7 State Finals

7 Mathematics 6 8. Bob can run from point to point B in minutes. Sue can bicycle from point to point B in minutes. Bob starts at point, runs to point B, and then runs back to point. Five minutes after Bob initially starts running, Sue starts bicycling from point, goes to point B, and then bicycles back to point. Do the two of them meet twice, and if they do, how long has Bob been running when they meet the second time? (Round your time to the nearest tenth of a minute.) a) No, they only meet once b) Yes, Bob has been running for 6.3 minutes c) Yes, Bob has been running for 7.5 minutes d) Yes, Bob has been running for 7.8 minutes e) Yes, Bob has been running for 8.5 minutes 9. Determine all solutions for for the following equation: (Round your answers to the nearest tenth.) a).3 b). c).,. d).3,.3 e) This equation has no real solutions. number minus it s reciprocal is equal to. What is the number? a) ± b) ± 3 c) ± 5 d) e) ±. Determine the equation of the fourth degree polynomial f ( ) if you know that crosses the y-ais at y 4, crosses the -ais at - and, and touches the -ais at. 4 3 a) f( ) b) f( ) c) f( ) d) ( ) 6 6 f e) f( ) f ( ) 7 State Finals

8 . Determine the length of the cross product B where 3i j + k and B i + j 3k (Round your answer to the nearest whole unit.) Mathematics 7 a) units b) 4 units c) 6 units d) 3 units e) 8 units 3. Toss two distinguishable dice. If the sum of the faces facing up is a prime number, you win $5, else, you lose $3. What is the epected value gain/<loss>? a) $</3> b) $</3> c) $/3 d) $/3 e) $/ 4. Determine the area of the ellipse that has the equation 5 + 9y + 8y + 4 a) 6 π square units b) 8 π square units c) 9 π square units d) 6 π square units e) 36 π square units 5. Determine the period of the function f( ) 5cos(4 ) + tan(3 ). a) 7 b) 6 π c) 4 π d) π e) π 6. For sets and B their universal U, what is I U? a) b) U c) d) B e) none of these 7. You are told that the locus of points that are solutions to the equation + y 4y + are put through the transformation T (, y) ( + y, y). What is the area enclosed by the points after the transformation? (Please round your answer to the nearest square unit.) a) 4 square units b) square units c) 8 square units d) 4 square units e) 57 square units 8. Find all solutions to the equation 3 n n n n. a) /3 b) 3/ c) 3/ d) /3 e) 3/, 3/ 9. couple has 7 kids. ssuming that the probability of having a boy is 5%, what is the probability that they will NOT have 3 or 4 boys? a) 3/64 b) 9/64 c) 34/64 d) 93/8 e) none of these 7 State Finals

9 Mathematics 8 3. boat is traveling at 5 mph parallel to the shore. t one point, a person on the boat notices she must look 3 degrees to the left to see a lighthouse on the shore. Thirty seconds later, she must look 5 degrees to the left to see the same lighthouse. ssuming a straight shoreline, how far is the boat from shore? (Round your answer to the nearest foot.) a) 6 feet b) 43 feet c) 54 feet d) 6 feet e) 739 feet 3. Determine all possible ways we can rearrange the letters of the word RERRNGE. (One possibility would be RRENGRE.) a) 5 b) 584 c) 648 d) 96 e) pump can fill a tub in 7 minutes. second pump is used to pump out water at a faster rate, and is turned on minutes after the first pump. If the tub is empty after 4 /3 minutes, how long will it take the second pump to empty a full tub when working alone? a) minutes b) 4 minutes c) 6 minutes d) 8 minutes e) 9 minutes 33. radioactive substance has a half-life of 8 days. Determine how long it should take for pounds of this substance to decay down to pound. (Round your answer, numerically, to the nearest whole day.) a) 93 days b) days c) days d) 4 days e) 8 days 34. Find all solutions for to the equation: e ssume C 4B and B 4C. + Be C a) b) C ± C 4B ln B ± ln B 4C c) d) C + C 4B ln B + ln B 4C e) C ln B 7 State Finals

10 Mathematics Find the slope of the line perpendicular to the curve y at /. a) /8 b) -3/4 c) 3/4 d) -4/3 e) 4/3 36. plane has a relative heading of 45 northeast and is traveling at a speed of 5 miles per hour. wind is blowing due east at mph. What is the true speed and direction of the plane to an observer on the ground? (Round your answer to the nearest degree and mph.) a) 3 mph 4 degrees north of east b) 55 mph 44 degrees north of east c) 85 mph 44 degrees north of east d) 55 mph 4 degrees north of east e) 65 mph 4 degrees north of east 37. The location of particle P at time t is given by the parametric form (5cos3t, 5sin3t). π Determine the length of the path that particle P takes as t goes from ttot 5π 5π 5π 5π 5π a) b) c) d) e) Find the angle between the following two lines: y (4/3) + and y -(/) + 3 a) arctan (/6) b) arctan (5.5) c) arctan (4/3) + arctan(/) d) arctan (4/3) - arctan(/) e) π / t a local ping pong tournament, eight contestants (my (f), Bob (m), Cheri (f), Dave (m), Erin (f), Fred (m), Gina (f), and Hal (m)) played three rounds of ping pong. No one faced the same opponent twice, and every game ended with one winner and one loser. Only one person won all three games (and therefore won the tournament), and only one person lost all three. The following is also true: I Hal won his first game against Bob, but lost to Dave and then to Cheri II Fred lost to Gina and then to my, but won his last game against Bob III Erin lost to women in rounds and, but beat a woman in round 3 IV The women had two more victories overall than the men V Two games each round had same-gender pairings VI Gina won her second match Who won the tournament? a) my b) Cheri c) Dave d) Gina e) Not enough info given 7 State Finals

11 Mathematics 4. farmer plans to build a rectangular fence around a garden, using a straight riverbank as one of the boarders (ie, he will use no fencing). If he has linear feet of fencing, what is the maimum square footage of his new garden? (Round to the nearest square foot.) a) 3,5 sq. ft. b) 4,75 sq. ft. c) 5,53 sq. ft. d) 6,7 sq. ft. e) none of these 7 State Finals

2005 Academic Challenge

2005 Academic Challenge 005 Academic Challenge MATHEMATICS TEST - STATE FINALS Mathematics Test Production Team Tom Ferry, Illinois State University Author/Team Coordinator Kevin Boyer, Illinois State University Author Sandra