Math 103 Set 4

Start reviewing for the first hour exam in Math 104 tonight. Get help (Math 103 and 104 staff, tutor room etc.) before the exam to firm up your understanding and problem solving ability.

1. Each group should discuss each of the problems below to identify relevant information and to analyze how it is to be solved. Make certain that everyone agrees with the analysis and understands it. Everyone then solves the problems individually, comparing answers and making corrections. Problem 1 is partially analyzed for you to help you to get started.
   1. The sum of two consecutive odd integers is 48, find the two numbers. In order to deal with this mathematically, we need names for the two integers in order to be able to manipulate them. Having done this, we need to examine the problem to find an equation (that we can solve) whose solution gives the required integers.

      Step 1: Give a name, say x, to the first integer.
      Step 2: Is there any information in the problem (consecutive) that will enable you to find an expression for the second integer in terms of x?
      Step 3: Is there any information in the problem (sum is 48) that will enable you to write down an equation involving the two integers?
      Step 4: Solve the equation and identify the two numbers.

   2. Three-tenths of a woman's monthly income goes to taxes. If she pays $900 a month in taxes, what is her monthly income?

   3. The width of a rectangle is 16 feet less than the length. Find the dimensions of the rectangle if the perimeter is 104 ft.

   4. It costs $80/day and $0.75/mile to rent a Porsche Carerra. If a three-day rental cost $786.00, how many miles were driven?

2. Group discussion: When is product of two numbers equal to 0? When is product of three numbers equal to 0? When is a product of four numbers equal to 0? When is a product of N numbers equal to 0?

   As a group, apply the above discussion to determine a sequence of steps that would solve problems 5 and 6. Do not write anything down! Each student then solves problems 5 and 6 by themselves, writing down their solutions, checking answers and making corrections as necessary.

   5. (z + 2)(z 8) = 0
      
   6. (x 2)(x + 5)(x 3) = 0

   As a group, determine what steps are needed to solve problems 7 and 8. Do not write anything down! Each student then solves problems 7 and 8 by themselves, writing down their solutions checking answers and making corrections as necessary.

   7. y⁴ = y²
      
   8. y³ + 6y + 8y = 0²

   Discuss how to solve the following equations without writing anything down. After everyone agrees on a sequence of steps that will lead to a solution, each student solves the problem by themselves, writing down their solutions, checking answers and making corrections as necessary.

   9. -3z² + 6 = 0
      
   10. 4y² - 8y = 0
   11. 9x2 = 18x
       

3. Discussion: Given a point P in the plane with coordinates (a,b), what is the geometric interpretation of the number a? The number b? If you are uncertain of the answer, check with a member of the MCM staff.
   12. Working individually, plot the following points using the above set of axes.
       P(-1, 2) Q(-6, -3) R(0, 4) S(-1, -3) T(-6, 4) U(10, 2)

   13. Referring to problem 13, examine each of the following pairs of points in sequence. For each pair, determine which of the two "lies to the right" of the other, and which of the two is "higher" than the other.
       {P,Q} {Q,S} {P,U} {R,T} {R,U} {T,S}

4. Group discussion: Consider the points P(a, b) and Q(c, d) in the plane. Can you determine which of the two points "lies to the right" of the other without plotting the points? Can you determine which of the two points is "higher" without plotting the points?
   14. If you moved the point G(1, 2) up 3 units, what are the coordinates of the new point? If you moved G(1, 2) down 4 units what are the coordinates of the new point?

   15. If c > 0, determine a relationship between the points P(a, b), Q(a,b+c) and R(a,b-c).

   16. Find a point which is 3 units to the right of ( 1, 2). Can you find two more points which are 3 units to the right of ( 1, 2)? How many such points are there?

   
   

1. A cube with edge 13 cm is completely submerged in a rectangular tank of water. If the tank is 50 cm long and 20 cm wide, how high will the water level rise?

Math 104 Text

Section 1.7 [40, 52, 56]
Section 1.8 [32,44]
Section 2.2 [6, 14, 26]

Remember: Try to finish every worksheet in class. If you are unable to, then see if you can get together with a group of students in Math 103 and work together outside of class. If that doesn't pan out then try to finish the worksheet by your self. Make certain that you see your TA or SCA to clarify any questions that you might have.

Strategy for Taking a Math Test

• If for any reason you arrive late, at either the exam or the make-up, speak to your TA or to one of the proctors and ask if it is possible to get some extra time at the end of the exam

• Get good nights sleep and avoid overly spicy foods. Eat a mild breakfast.

• Since you are taking an evening exam, try to relax sometime during the day. If you are prone to sugar depletion, bring some M&M's (or alternate of your choice) with you to the exam.

• Problems on Math tests are rarely organized in order of increasing difficulty. One of the most important things for you to do, is to make sure that you start with problems that you find easy to do. Here's one strategy that can accomplish this. When you first get your exam, take a minute to look it over. Place a check next to the problems that you feel most comfortable with. Start working on these problems first. You will find that you will tend to relax more if you begin the test with problems that you are able to do. When you've completed these, look over the remaining problems and try to determine which ones you are more apt to get right. Do those next. Continue in this fashion until you complete the test or time runs out. Re-read this paragraph!

• Try to write neatly and carefully. That way, if you happen to finish the exam early, it will be easier to check your work in the remaining time.

• Make certain that you follow the directions for each problem. Do not simplify an answer unless there is a reason to do so. When in doubt, ask a proctor at the exam.

• If you do not understand a problem, it doesn't hurt to ask a proctor if she/he could clarify the problem.

Good Luck!