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CHAPTER RESOURCES : LINKING CONCEPTS PROBLEMS
Mark Dugopolski
The problems given here may be more challenging or may require more ingenuity than those found in the text. These
problems are arranged by chapter, but many of them can be solved with a variety of techniques. You don't necessarily
have to wait until you've completed a certain chapter to try a problem. A graphing calculator will be helpful or
necessary on some problems, but an algebraic solution should be given when possible. These problems could be used
as bonus problems for individuals or groups. The answers to these problems are not posted on this Web site. If
you're submitting solutions of these problems to your instructor, be sure to explain every detail of your reasoning.
Try typing your work with a scientific word processor - it's good practice and will make a good impression on your
instructor. I'll keep looking for new problems and post them here periodically.
Chapter P [ TOP ]
- Consider the equation (x+1)(y+1)=2xy. Find two pairs of positive integers that satisfy this equation and explain
how you found them.
- Consider the equation (x+1)(y+1)(z+1)=2xyz. Find some solutions to this equation where x, y, and z are positive
integers and explain how you found them.
Chapter 1 [ TOP ]
- A rectangular sheet of paper measuring a by b is black on one side and white on the other. It is lying on a
table with a black side up.
a. If you fold the paper on the diagonal of the rectangle, then what is the area of the black triangle that
you now see?
b. Find the least upper bound for the area of the black triangle.
Chapter 2 [ TOP ]
- Consider the circle of radius 5 centered at the origin. Find the point on this circle that is closest to the
point (7,9).
- Consider the parabola y = x2. Find the point on the parabola that is closest to the point (3,1).
Chapter 3 [ TOP ]
- A rectangular piece of paper measuring a by b is placed in the first quadrant so that its vertices are (0,0),
(0,b), (a,0), and (a,b).
a. If the paper is folded so that the corner at (a,b) coincide with (0,0), then find the equation of the crease.
b. If the paper is folded so that the corner at (a,b) coincide with (0,b/2), then find the equation of the crease.
- Find the distance from the point (2,3) to the line y = 4 x - 7.
- Imagine that a mirror is placed on the line y = 3 x - 2 perpendicular to the x y plane. An ant is sitting at
the point (3,1). Find the coordinates of the ant's reflection in the mirror.
Chapter 4 [ TOP ]
- Show that loga(b) - logb(a) = 1
- Show that
Chapter 5 [ TOP ]
- Consider the triangle with vertices (0,0), (0,a), and (b,c) where a, b, and c are positive numbers.
a. For each vertex, write the equation of the line through that vertex and the midpoint of the opposite side
of the triangle.
b. Find the intersection of the three lines that you found in part (a).
- Consider the triangle with vertices (0,0), (0,a), and (b,c) where a, b, and c are positive numbers.
a. For each side of the triangle, write the equation of the line that is perpendicular to that side and passes
through the midpoint of the side.
b. Find the intersection of the three lines that you found in part (a).
- Select any three ordered pairs. Find the equation of the circle that goes through your ordered pairs. Under
what conditions will there fail to be a circle passing through your ordered pairs.
Chapter 6 [ TOP ]
- Consider the triangle with vertices (0,0), (0,a), and (b,c) where a, b, and c are positive numbers.
a. For each vertex, write the equation of the line that contains that vertex and is perpendicular to the opposite
side of the triangle.
b. Find the intersection of the three lines that you found in part (a).
- While driving along an interstate highway at a constant speed (cruise control), Hector observed a mile marker
containing a three-digit number in which the tens digit was zero. Exactly one hour later he observed a mile marker
containing a two-digit number in which the digits were first and last digits (not necessarily in that order) of
the previously observed mile marker. Exactly one hour later he observed a third mile marker containing another
two-digit number in which the digits were the same two digits that he observed one hour earlier. What was his speed?
- Find a matrix A such that A5 = 0, but A # 0.
Chapter 7 [ TOP ]
- Select any two points on a hyperbola. These two points are opposite vertices of a parallelogram formed by drawing
lines through these points running parallel to the asymptotes of the hyperbola. Prove that the line through the
other two opposite vertices of the parallelogram goes through the center of the hyperbola.
- Prove that if the asymptotes of a hyperbola and a single point on the hyperbola are given, then every other
point on the hyperbola can be located with a straight edge and a compass. See the previous problem.
Chapter 8 [ TOP ]
- On the game show Let's Make a Deal, host Monte Hall shows a contestant 3 unopened curtains. The curtains hide
a new BMW and two zonks and Monte knows the location of the prizes. A contestant must select a curtain. Then Monte
opens one of the unselected curtains to show the contestant a zonk. Monte then gives the contestant the opportunity
to trade her selected curtain for the other unopened curtain. What should she do?
- A new family is moving into the house next door, and you know that they have only two kids. You see the mover
unloading a child's size girl's bicycle. Taking that information into account, what is the probability that the
family has two girls?
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