Extra Credit Problems
Second Six Weeks

Instructions:
  1.  Problems may be submitted any time during the six weeks, but they MUST be turned in by the due date.
        (Due Date:  October 30, 2007)
  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.   A builder is purchasing a rectangular plot of land with frontage on a road for the purpose of
      constructing a rectangular warehouse. Its floor area must be 300,000 square feet. Local
      building codes require that the building be set back 40 feet from the road and that there be
      empty buffer strips of land 25 feet wide on the sides and 20 feet wide in the back. Find the
      overall dimensions of the parcel of land and building which will minimize the total area of the
      land parcel that the builder must purchase.
      (Problem taken from Problems in Calculus published by J. Weston Walch--1985)

2.   A driver on a desert road discovers a hole in the gas tank leaking gas at the constant rate of 4
      gallons per hour. This driver, having no way to plug the hole, decides to drive for as long as
      the gas supply allows. The gauge reading indicates the tank is three-fourths full, which means
      that the tank contains 14 gallons. The car consumes gas at the rate of 18 miles per gallon at
      40 mph.  For each 5 mph below 40 mph add one-half mile per gallon to this rate; for each 5
      mph above 40 mph, subtract one mile per gallon from this rate. If the driver chooses the best
      constant speed in order to get the maximum driving distance, find the maximum distance that
      the 14 gallons will allow. Assume that gas consumption is a continuous function of speed.
      (Problem taken from Problems in Calculus published by J. Weston Walch--1985)

3.   Water is being pumped at a constant but unknown rate from a conical tank 10 meters high
       and 8 meters across. At noon the height of water is 7 meters, and you observe that the water
       level drops 3 cm over the next minute. Estimate when the tank will be pumped empty.
       (Taken from Probe)

4.  Analyze the graph of y = cos x + cos 2x over the interval [0,2p].  Find the x-intercepts,
     y-intercepts, local maximums, local minimums, where the graph increases, where the graph
    decreases, where the graph is concave up, where the graph is concave down, inflection points,
    and asymptotes.  Then sketch it as accurately as possible.  Show all your analyses.  Answers
    may not come from the calculator.

5.  Determine the maximum area of a rectangle with one side on the x-axis and the opposite
     corners touching the parabola y = -x² + 9.

6.  Find the lengths of the sides of the isosceles triangle with perimeter 12 and maximum area.

7. One ship, A, is sailing due south at 16 knots and a second ship, B, initially 32 nautical miles
    south of A, is sailing due east at 12 knots.
    a. At what rate are they approaching or separating at the end of one hour?
    b. When do they cease to approach one another and how far apart are they at this time? What
         is the significance of this distance? 
    (Taken from Problems in Calculus, J. Weston Walch, Publisher, 1985)