Finite Math 107 A
Exam 2 Practice Problems
Tom Linton, Spring 2004

1. You run a food stand at tulip time and have decided to sell buffalo wings, but are unsure of the price you should set for the spicy treats. You do know the following pieces of information however:
• If you sell wings for 50 cents each, customers will demand 400 wings and you will supply 300 wings.
• If you sell a wing for 75 cents, customers will buy 350 of them, while you will be willing to supply 500 wings.
You are interested in setting the price of a buffalo wing at the equilibrium value (where the supply and demand curves intersect). Assume that both the supply and demand curves are linear for this situation.
1. On the axes below, mark the 2 points referred to above, which lie on the demand curve, with a "dot".
   See plot.
   
2. On the axes below, mark the 2 points referred to above, which lie on the supply curve, with an "x".
   See plot.
   
3. Draw in the supply and demand curves (connect the dots and x's) and use your drawing to estimate the appropriate price for a buffalo wing.
   See plot, the equilibrium is about at q = 375 wings, p = 60 cents.
   
4. Find the linear equation for the demand price, D(q). What is the demand price if q = 200?
   D(q) = -q / 2 + 250, D(200) = 150 cents.
   
5. Find the linear equation for the supply price, S(q). How many wings will you supply at a price of 40 cents per wing?
   S(q) = q / 8 + 25 / 2. If p = 40, q = 220 wings.
   
6. Using your graphing calculator, find the equilibrium point, (q, p). Was your estimate for the price of a buffalo wing from part (c) close to correct?
   The equilibrium is at q = 300, p = 60. Set Y1 = D(X), Y2 = S(X), set the window to that in the plot above and use [CALC][5:INTERSECT].
   
7. What is the numeric value of the p-intercept for the demand curve? Interpret this value.
   Setting q = 0 for the demand curve gives p = 250 cents. At a price of $2.50 a wing, no one will buy any wings.
   
8. What is the numeric value of the q-intercept for the supply curve? Interpret this value.
   Setting p = 0 for the supply curve gives q = -100 wings. At a price of $0 a wing, the supply would be negative 100 wings (a meaningless value).
   
9. If we denote both the supply price, S(q), and the demand price, D(q), as simply p (in cents), the equilibrium point corresponds to the solution of the system 2p + q = 500 and 8p - q = 100. Solve this system using elimination.
   Adding the 2 equations gives 10p = 600, so p = 60 cents. Plugging p = 60 into the first equation gives q = 380 wings.
   
10. Re-write the system given (before it is solved) in the last part in matrix form A*X = B, where X = .
    =
    
11. Find the matrix A^-1.
    A^-1 =
    
12. Verify (by matrix multiplication) that A^-1*B also gives the equilibrium point.
    *=
    
2. Sol's deli sells three sizes of pastrami sandwiches, small, medium, and large. A small sandwich sells for $3.00 and contains 4 ounces of pastrami. A medium sandwich sells for $3.50 and contains 8 ounces of meat, while the large goes for $4.00, with 11 ounces of pastrami. Sol receives an order for 58 pastrami sandwiches that require 26 pounds, 2 ounces of meat (there are 16 ounces in a pound), for a total cost of $199.00.
1. Define the variables and set up a system of equations to determine the number of sandwiches of each size in the order.
   x = number of small sandwiches, y = number of medium sandwiches, z = number of large sandwiches.
   x + y + z = 58
   3x + 3.5y + 4z = 199
   4x + 8y + 11z = 418
   
2. If you had to solve this system of linear equations by elimination, what would your first 2 steps be? Why would you select these steps?
   There are many options, but I would subtract 3 times equation 1 from equation 2, and subtract 4 times equation 1 from equation 3. This owould eliminate x from the system of equations.
   
3. If f(x) = 2x + 6, and g(x) = 3x + 9, where do the graphs of f(x) and g(x) intersect?
   2x + 6 = 3x + 9, gives x = -3, and plugging that into either function gives y = 0, so they intersect at (-3,0).
   
4. Each augmented matrix below represents a system of linear equations (in the standard variables x,y, etc.). For each matrix, decide if the system has a unique solution (if so, give this solution), no solutions, or infinitely many solutions (if so, find three specific solutions and also describe all solutions with a parameter).
1. No solutions.
   
2. Infinitely many solutions. Let z be any number, then x = 4 + 2z, y = 0 is a solution. In particular, setting z = 0, 1, and -1 gives the three solutions (4,0,0), (6,0,1), and (2,0,-1).
   
3. A single solution, x = -1 and y = 4.
   
5. The Pella Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange and 1 quart of kiwi juice per gallon. There are 4 quarts in a gallon. Each day the company has 800 quarts of pineapple juice, 650 quarts of orange juice, and 350 quarts of kiwi juice available. How many gallons of each blend should the company make each day if it wants to use up all of the supplies?
   Let x = number gallons PineOrange, y = number gallons PineKiwi, and z = number gallons OrangeKiwi, then we have:
   (pineapple juice): 2x + 3y + 0z = 800
   (orange juice): 2x + 0y + 3z = 650
   (kiwi juice): 0x + y + z = 350. Putting the augmented matrix into the calculator and using rref gives x = 100 gallons of PineOrange, y = 200 gallons of PineKiwi, and z = 150 gallons of OrangeKiwi.
   
6. Consider the matricies A = and B = .
1. What is a_2,3? 4
   How about b_3,1? 7
   
2. What is B^T?
   B^T= 
   
3. Find the matrix product A*B.
   A*B =
   
4. Find A^-1.
   A inverse is B.
   
5. Find B^-1.
   B inverse is A.
   
6. Solve the system A*=
   Multiply by A^-1 (i.e. B) on the left to get x = 5, y = 2, z = 5.
   
7. Solve the system B* =
   Multiply by B^-1 (i.e. A) on the left to get x = -3, y = 7, z = 9.
   
8. Solve the system A*= . Your answer will involve the unknown constant a.
   Multiply by A^-1 (i.e. B) on the left to get x = 5+8a, y = 3+3a, z = 4+7a.
   
7. Consider the system of linear equations: 2x - 2y + 3z = 7, -2x + y - 3z = 2, -2x - y + 2z = -3.
1. Rewrite the system as an augmented matrix.
   
   
2. What are the dimensions of your augmented matrix?
   3 by 4.
   
3. Carry out the row operation R₁ + R₂ -> R₂ on your augmented matrix.
   
   
4. Carry out the row operation R₁ + R₃ -> R₃ on your NEW augmented matrix.
   
   
5. What row operations would you use to make the second column of your new augmented matrix look like ?
   -1*R2 -> R2, then 2*R2 + R1 -> R1, and then 3*R2 + R3 -> R3 to get