Extra Credit Problems
Fifth Six Weeks

Instructions:
  1.  Problems may be submitted any time during the six weeks, but they MUST be turned by the due date.
        (Due Date:  April 4, 2008)
  2.  Your work must accompany each problem and the problem must be stated or printed out.
        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. Solve the system:  x2 + 3y2 = 3
    2.                           3x2 - xy = 6
     
  2. A standard deck of 52 cards is shuffled and the cards are dealt face up one at a time until an ace appears.
    3.     a.  What is the probability that the first ace apears on the third card?
        b.  Show that the probability of getting the first ace on or before the ninth card is greater
             than 50%.
     
  3. Of the 415 girls at Gorham High School last year, 100 played fall sports, 98 played winter
    4. sports, and 96 played spring sports.  Twenty-two girls played sports all three seasons while
    40 played only in fall, 47 only in winter, and 33 only in spring.
       a.  How many girls played fall and winter sports but not a spring sport
       b.  How many girls did not play sports in any of the three seasons?
     
  4. Prove that n5 - n is always divisible by 10, for all positive integers.
    5.  
  5. Prove that if n is an integer satisfying n4 + 4n3 + 3n2 + n + 4000 = 0, then n is even.
    6.  
  6. Prove that

    7. is a composite number for n > 2.

  7. Prove that between any two consecutive multiples of 7 there are at least two multiples of 3.
    8.  
  8. Prove the formula for summing fourth powers of integers found on page 690 of your textbook.  

    9. Because no one was able to solve problems 2,3, and 4 from the 4th 6 weeks extra credit, I have put them on this six weeks extra credit and you may now work them  for 5 points each. Spend a little time with them!
     

  9. A fuel tank has a cross section whose shape is a 2 m by 2 m square capped at the top and bottom 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.
    10.  
  10. 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.

  11. From the southeast corner of the cemetery on Burnham Road, proceed S 78o W for
    12. 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.

14.  Simplify each of the following:

  1. log232
  2. (-3)log-35
  3. 7 log7 (a + 3)
  4. log168
  5. \/2 log43
15.  Radioactive Iodine with a half-life of 8.1 days is used to determine if people have a thyroid
       deficiency. An amount, N, of the Iodine is injected into the blood stream and is absorbed by
       a healthy thyroid gland. By measuring the thyroid's radioactivity at various later times, it is
       possible to tell whether or not the thyroid is functioning normally. Express in terms of N, the
       amount of radioactive Iodine which should be present in a healthy thyroid gland 6 days after
       it was injected into the blood stream.

16.  Given a population of 10,000,000 and an annual growth rate of 3%, how long will it take
       this population to double? Will the population quadruple in twice this doubling time? What is
       the size of the population in triple the doubling time?

17.  For Martha's birthday present her parents presented her with a bank account with $16,000
       in it. When she was born, her parents had placed $2000 in a savings account which was
       guaranteed to double every eight years. How old is Martha?

18.  What exponential function of the form y = aebx goes through the points (3,10) and (6,50)?

19.  The surface area of the Earth is approximately 197 million square miles, including oceans.
        If the population of the Earth is approximately 4.9 billion people, how much room does
        each person have? Suppose the doubling time for the world's population is approximately 41
        years, when did each person have 1 square mile?

20.  A microcomputer system which currently sells for $1100 sold in 1982 for $1850. Assume
       that the cost continues to decrease and that this decrease is exponential.

  1. Derive an equation for the cost based on time since 1982.
  2. Use this equation to predict the cost at the end of one more year.
  3. When will the price fall to $500?
21.  The woods north of town have two types of rabbits. The light brown, fuzzy ones have a
       current population of 300 and doubles the population every 1.4 years. The second kind of
       rabbit is dark brown with pink ears. Its initial population is 1560 and the population doubles
       every 2.5 years. How long before the colonies are the same size?