Extra Credit Problems
Third Quarter

Instructions:
  1.  Problems may be submitted any time during the quarter, but they MUST be turned in by the due date.
        (Due Date:  February 28, 2011)
  2.  Your work must accompany each problem.  No credit will be given for just an answer.
  3.  Parts of problems are worth anywhere from 1 - 3 points depending on level of difficulty.
       Credit is given only for correct work and answers. The whole part must be correct to
       receive credit.
  4.  You may do as many or as few problems as you desire. Try to keep them in order.

Problems:
  1.  At time zero, a weight hanging at rest from a spring is pulled down a distance of 8 cm and
    2.  released. The weight oscillates up and down and makes 1 complete cycle every 2 seconds.
    Write an equation to model the situation, and state the amplitude, period, and frequency.
     
  2. Graph: y = sec 2(x - p) without a calculator.

    3.  
  3. Graph y = 2 tan [3(x - p/4)] + 1, showing all your analyses.  Do not use a calculator.

    4.  
  4. Solve for x:  sin x = tan x  for   -360o < x < 360o

    5.  
  5. Solve for x:  2 cos2x + cos x = 0  for  0 < x < 2p

    6.  
  6. Solve for x:  sin x < cos x  for  0 < x < 2p

    7.  
  7. Verify (or prove):  verify

    8.  
  8. A surveyor determines that the distances from a house to each of the endpoints
    9. of the diameter of a small circular lake are 1000 m and 2500 m.  If the angle
    between these two sides of the resulting triangle is 72o, what is the diameter of
    the lake? Draw a sketch and find the area of the lake.
     
  9. How many triangles are there with an angle of 58o, adjacent side 18, and
    10. opposite side 17?   If there are any, solve for them.
     
  10. A boat A sights the lighthouse B in the direction N65oE and the spire of a
    11. church C in the direction S75oE. According to the map, B is 7 miles from C
    in the direction N30oW. In order for A to avoid running aground, find the
    bearing it should keep to pass B at 4 miles distance.
     
  11. Solve for 0 < x < 2p:      tan (3p x) - 4 = 1   

  12. Use DeMoivre's Theorem to derive an identity for sin 3q  in terms of cos q and sin q.

  13. A fuel tank has a cross section whose shape is a 2 m by 2 m square capped at the top and bottom
    14. by semicircles.  Use a computer or graphing calculator to determine how to mark a measuring rod
    to show that the tank is only 10% full.
     
  14. A goat is tethered to a stake at the edge of a circular field with radius 1 unit.  Use a computer or graphing calculator to determine how long the rope should be so that the goat can graze over
    half the field.

  15. From the southeast corner of the cemetery on Burnham Road, proceed S 78o W for 250 m
    along the southern boundary of the cemetery until a granite post is reached, then
    S 15o E for 180 m to Allard Road, then N 78o E along Allard Road until it intersects
    Burnham Road, and finally N 30o E along Burnham Road back to the starting point.
    Find the area of this plot of land.

  16. Evaluate sin [Tan-1(1/2) + Tan-1(1/3)] without using a calculator or tables.

  17. Verify that 4 Tan-1(1/5) - Tan-1(1/239) = p/4   
    (Hint:  Let a = Tan-1(1/5) and let b = Tan-1(1/239).  Then find tan (4a - b).)

  18. Solve for x for 0 <  x < 2p. 
    a.  sin 3x = sin 5x + sin x
    b.  Tan-12x = Sin-1x

  19. Find the coordinates of the point 4/5 of the way from A(7,-2) to B(2,8).

  20. Let x and y be positive integers such that x + y < 13 and 3x + y < 24. 
    Find the maximum value of 4x + y

  21. Graph |x| + |y| = 2 without a calculator.

  22. Graph |2x - 3y| < 6 without a calculator.

    (Numbers 12-22 were taken from Advanced Mathematics by Richard G. Brown, Houghton Mifflin, 1992)

  23. Solve the system:  x2 + 3y2 = 3
      24.                   3x2 - xy = 6
  24. Prove that n5 - n is always divisible by 10, for all positive integers.

  25. Prove that if n is an integer satisfying n4 + 4n3 + 3n2 + n + 4000 = 0, then n is even.

  26. Prove that                     is a composite number for n > 2. 

  27. Prove that between any two consecutive multiples of 7 there are at least two multiples of 3.

  28. Prove the formula for summing fourth powers of integers found on page 690 of your textbook.