M360M/396C, Fall 05

PROBLEM SETS

#1

NOTE: One purpose of this first problem set is to get you started with some problems that have extra, missing, incomplete, misleading, or ambiguous information, or that have more than one or no solution, or that can't be answered precisely -- and to give you reason to read problems carefully, and look before you leap. Forewarned is forearmed. (But please don't start thinking that every problem I give you is a "trick question"!)

1. Way back when, a handcar set out from Chicago to Detroit at an average of 10 miles per hour. Four hours later a freight train started a run from Detroit to Chicago at an average speed of 20 miles per hour. Assuming the rail distance between the two cities is 300 miles, which was closer to Chicago when they meet?

2. Consider rectangles which have one vertex at the origin, a different vertex on the y-axis, a third on the x-axis, and th fouth vertex on the graph of y = 4 - x². Of all such rectangles, find the dimensions of the one with the largest area.

3. (This is a variation of an old problem attributed to Charles Lutwidge Dodgson, who wrote the Alice in Wonderland stories using the pseudonym Lewis Carroll)
   Jack and Jill walked along a level road, up the hill, back (along the same path) down the hill, and back along the same level road to home. They started out at 3 p.m. and arrived home at 9 p.m. the same day. (<-- Added 9/19/05) Their speed was four miles an hour on the level, three miles an hour uphill, and six miles an hour downhill.
   a. How far did they walk in all (level, up, down, level)?
   b. What was their average speed for the whole trip?
    c. How closely can you find the time when they arrived at the top of the hill? (For example, can you give an interval of an hour containing the time they arrived at the top? An interval of a half hour? Or can you tell the exact time? )

#2

1. You turn on an oven and set the temperature to 350°. Sketch a graph of the temperature in the oven (say, at a temperature probe) versus time since you turned on the oven. Explain (in a fair amount of detail) why your graph looks like it does.

2. During the hottest months in Austin, the angle which the noonday sun (shining from the south) makes with the horizontal varies from 70° to 80°. During the coldest months, this angle ranges from 36° to 39°. A house has a south facing window 6 ft. in height (measured from sill to top of window) which starts 5 ft. above the ground. (That is, the sill is 5 ft above the ground). There is a horizontal overhang placed 2 ft. above the top of the window.
    a. How long must the overhang be to shade the window completely at noontime during the hottest months?
    b. What is the longest the overhang could be to avoid shading any part of the window during the coldest months?
    c. How could you design an overhang to achieve both the goal in (a) and the goal in (b)? (You may change the conditions of being 2 ft above the window and of being horizontal.)

3. Find an equation (in simplified form) for the set of all points equidistant from the line y = 1 and the point (-2,3)

4. You have eight blocks of ice, each 30 cm. X 30 cm. X 30 cm.
    a. How should you stack them if you want them to melt as slowly as possible?
    b. How should you stack them if you want them to melt as quickly as possible? (Constraint: Each block of ice must have at least one side up against the entire side of another block of ice. In other words, you can't spread them all out separately.)
(As always, you must explain why your answer is the best possible. )

#3

1. a. Find all possible real numbers m so that the graphs of the equations x + my = 0 and x = y² have exactly one point of intersection.

b. Find all possible real numbers m so that the graphs of the equations x + my = 0 and x = y² have no points of intersection.

c. Find all possible real numbers m so that the graphs of the equations x + my = 0 and x = y² have exactly two points of intersection.

2. Take a positive integer; square it, then add 1. Can the result ever be (evenly) divisible by 6?

3. What angle do the hands of a clock make at time 7:38?

4. Two players play the following game: Thirteen markers are arranged in a circle. Each player in turn removes either one marker or two markers that started out next to each other. The player who takes the last marker wins. Is there a winning strategy for one of the players? If so, for which one, and what strategy? (A winning strategy is a method of deciding how to play your moves that will always lead you to win, no matter what the other player does.)

#4

1. If n is a positive integer, can n² + n ever be the square of an integer? Why or why not? What if n is a negative integer?

2. A satellite is 600 kilometers above the earth. How far (in kilometers) along the earth's surface could a camera on the satellite "see"? (Think carefully about what this last sentence means. There is more than one interpretation that fits the question, but all interpretations that do fit are pretty much the same problem. In case you need it, the radius of the earth is 6370 kilometers.)

3. The game of toe-tac-tic has the same rules as the game tic-tac-toe, with one exception: the first player with three markers in a row loses. Is there a strategy which one player can follow to avoid being beaten, no matter what the other player does? If so, describe the strategy precisely and prove that it works. If not, prove why not.

4. a. For what values of a does the system of equations

        x² - y² = 0
        (x - a)² + y² = 1

have no solutions?
b. For what values of a does the system have exactly 1 solution?
c. For what values of a does it have exactly 2 solutions?
d. Exactly 3?
e. Exactly 4?
   f. Exactly 5?

#5

(Passed out in class Tuesday, October 25)

#6

(Passed out in class Thursday, November 10)

#7

(Passed out in class Thursday, November 17.)