Practice Exam
- Page ID
- 34522
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Exercise \(\PageIndex{1}\)
What will the future value of $5000 be in 5 years at 8% compounded quarterly?
- Answer
-
7,429.74
Exercise \(\PageIndex{2}\)
A bank pays 8% compounded quarterly, what is the effective interest rate?
- Answer
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8.24%
Exercise \(\PageIndex{3}\)
Each year a sum of $2200 is placed in an IRA account paying 12%. After 10 years what will the final amount be?
- Answer
-
$38,607.22
Exercise \(\PageIndex{4}\)
If $5000 is invested at 8% compounded daily for 3 years, find the final amount.
- Answer
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$6,356.08
Exercise \(\PageIndex{5}\)
Find the present value of $200 per month at 12% for 4 years.
- Answer
-
$7,594.79
Exercise \(\PageIndex{6}\)
A $12,000 car loan is amortized at 6% over 6 years. Find the monthly payment.
- Answer
-
$198.87
Exercise \(\PageIndex{7}\)
In problem \(\PageIndex{6}\), what will the balance of the loan be after 4 years?
- Answer
-
$4,487.08
Exercise \(\PageIndex{8}\)
If in problem \(\PageIndex{6}\), the loan is amortized over 3 years, what will the monthly payment be?
- Answer
-
$365.06
Exercise \(\PageIndex{9}\)
A $24,000 car has a useful life of 6 years. It can be leased for $400 per month. If the current interest rate is 6%, is it better to lease or to buy?
- Answer
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buy
Exercise \(\PageIndex{10}\)
What monthly payment will amount to $10,000 in 5 years at 11%?
- Answer
-
$125.75
Exercise \(\PageIndex{11}\)
A bond has a face value of $1000 and is due in 5 years. It pays $45 interest every 6 months. If the current interest rate is 12%, what is the fair market value of the bond?
- Answer
-
\(559.39 + 331.20 = 889.59\)
Exercise \(\PageIndex{12}\)
A company produces solar panels. The total cost of 20 panels is $2100, and the cost of 60 panels is $3900. Express the cost \(y\) in terms of the number of panels \(x\).
- Answer
-
\(y = 45x + 1200\)
For problems 13, 14 and 15, the final tableaux is given below
Exercise \(\PageIndex{13}\)
If the initial problem is a maximization problem, what is the maximum value?
- Answer
-
2500
Exercise \(\PageIndex{14}\)
If the initial problem is a maximization one, at what point does the maximum value occur?
- Answer
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\((200, 400, 100)\)
Exercise \(\PageIndex{15}\)
If the initial problem is a minimization one, at what point does the minimum value occur?
- Answer
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\((75,5,90\)
Exercise \(\PageIndex{16}\)
At a price of $3, 100 units are demanded of a product. At a price of $5, 50 units are demanded. If \(x\) is the price and \(D\) the number of units demanded, write the demand equation.
- Answer
-
\(D = -25x + 175\)
For problems 17 and 18, use the following system of equations.
\[\begin{array}{l}
2 x+4 y=10 \\
3 x+6 y=15 \\
5 x+10 y=27
\end{array} \nonumber \]
Exercise \(\PageIndex{17}\)
If \(x=1\), what is the value of \(y\)?
- Answer
-
No solution
Exercise \(\PageIndex{18}\)
How many solutions does this system have?
- Answer
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None
Problems 19 - 22 refer to the maximization problem below to be done using the simplex method.
A company makes two types of widgets, Regular and Deluxe. Each type of widget requires the use of three machines for its production. The Regular widget requires three hours on machine I, one hour on machine II, and one hour on machine III, and sells for $10. The Deluxe widget requires one hour on machine I, two hours on machine II, and one hour on machine III, and sells for $20. The maximum number hours available on Machines I, II, and III are 120, 100, and 40, respectively.
Exercise \(\PageIndex{19}\)
What are the coefficients of the objective function?
- Answer
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10, 20
Exercise \(\PageIndex{20}\)
What is the constraint imposed by Machine III?
- Answer
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\(x_1 + x_2 \leq 40\)
Exercise \(\PageIndex{21}\)
In solving this problem using the simplex method, how many slack variables are needed?
- Answer
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3
Exercise \(\PageIndex{22}\)
After the first full pivot operation, what is the current revenue?
- Answer
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800
Exercise \(\PageIndex{23}\)
A firm produces floppies at a variable cost of $1.20 per disk and a fixed cost of $1800. If the disk sells for $3 each, find the break-even point.
- Answer
-
\((1000, 3000)\)
Problems 24 - 25 refer to the following minimization problem:
A diet must contain at least 60 units of protein, and 30 units of fat. Food A provides 2 grams of protein and 4 grams of fat, and costs 30 cents. Food B provides 6 grams of protein and 2 grams of fat, and costs 20 cents.
Exercise \(\PageIndex{24}\)
Graph the constraints, and shade the feasibility region.
- Answer
-
Graph
Exercise \(\PageIndex{25}\)
Write the cost function.
- Answer
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\(C=.30x+.20y\)
For problems 26 and 27, the supply and demand equations are given as follows:
\(S=2/3x-100\)
\(D=-4/3x+500\)
where \(x\) is the price.
Exercise \(\PageIndex{26}\)
Find the equilibrium price.
- Answer
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300
Exercise \(\PageIndex{27}\)
How many items will be demanded at that price?
- Answer
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100
Exercise \(\PageIndex{28}\)
If there are 5 people in a room, what is the probability that no two have the same birthday?
- Answer
-
.97286
Exercise \(\PageIndex{29}\)
\(A\) and \(B\) are mutually exclusive, \(P(A)=.4\), \(P(B)=.5\) find \(P(A \text{ and } B)\)
- Answer
-
0
Exercise \(\PageIndex{30}\)
\(A\) and \(B\) are independent. \(P(A)=.4\), \(P(A \text{ and } B)=.24\), find \(P(B)\).
- Answer
-
.6
Exercise \(\PageIndex{31}\)
What is the probability of getting 3 heads if a coin is tossed 5 times?
- Answer
-
\(.3125\)
Exercise \(\PageIndex{32}\)
If \(P(A)=.5\), \(P(B)=.4\) and \(P(A \text{ and }B)=.2\), find \(P(A \text{ or } B)\).
- Answer
-
.7
Exercise \(\PageIndex{33}\)
How many different ways can two boys and three girls be chosen from a total of 6 boys and 8 girls?
- Answer
-
840
Problems 34 - 36 refer to the following information.
Companies \(A\), \(B\), and \(C\) produce 15%, 40%, and 45% respectively of the major appliances in an area. One percent of company \(A\) appliances, 2% of company \(B\) appliances, and 3% of company \(C\) appliances require service within the first year.
Exercise \(\PageIndex{34}\)
What is the probability that an appliance chosen at random is defective?
- Answer
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.023
Exercise \(\PageIndex{35}\)
If an appliance is chosen at random and found to be defective, what is the probability that it came from company \(B\)?
- Answer
-
.3478
Exercise \(\PageIndex{36}\)
Suppose it was manufactured by company B\(B\), what is the probability it is a defective appliance?
- Answer
-
.02
Exercise \(\PageIndex{37}\)
On 30% of his quizzes a student receives a score of 8, and on 70% his score is 9, what is his average?
- Answer
-
8.7
Problems 38 - 40 refer to the following.
An urn contains 3 red, 4 white and 5 blue marbles, and two marbles are drawn at random.
Exercise \(\PageIndex{38}\)
What is the chance of getting a blue marble on the second draw given that a red has been drawn on the first?
- Answer
-
5/11
Exercise \(\PageIndex{39}\)
What is the probability of obtaining one white and one other marble?
- Answer
-
.4848
Exercise \(\PageIndex{40}\)
What is the probability of obtaining at least one white marble?
- Answer
-
.5758
For problems 41 - 43, consider the following transition matrix giving the probabilities for the next purchase of Tide and Brand \(X\).
| Next purchase | |||
| Tide | Brand \(X\) | ||
| Present | Tide | .8 | .2 |
| Purchase | Brand \(X\) | .4 | .6 |
Exercise \(\PageIndex{41}\)
What percentage of the Tide people will buy Brand \(X\) next month?
- Answer
-
.2
Exercise \(\PageIndex{42}\)
If the original share of the market is (.25.75), what will the share be two months from now?
- Answer
-
[.6 .4]
Exercise \(\PageIndex{43}\)
What will the long term share of the market be?
- Answer
-
\(\left[\begin{array}{ll}
2 / 3 & 1 / 3
\end{array}\right]\)
For problems 44 - 46, consider the following transition matrix for an absorbing Markov Chain.
Exercise \(\PageIndex{44}\)
Identify the absorbing states
- Answer
-
1 and 4
Exercise \(\PageIndex{45}\)
Write the solution matrix
- Answer
-
Exercise \(\PageIndex{46}\)
Find the probability of ending in state 4, given one started in state 2.
- Answer
-
.4
Exercise \(\PageIndex{47}\)
Given the \(3 \times 3\) game \(\left[\begin{array}{ccc}
2 & 3 & 3 \\
1 & 0 & -1 \\
0 & 0 & 4
\end{array}\right]\), find the optimal strategy for the column player.
- Answer
-
\(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right]\)
For problems 48 - 50, consider the following \(2 \times 2\) payoff matrix.
\[ \left[\begin{array}{cc}
-1 & 0 \\
1 / 4 & -1 / 4
\end{array}\right] \nonumber \]
Exercise \(\PageIndex{48}\)
Find the row player's optimal strategy.
- Answer
-
\(\left[\begin{array}{ll}
1 / 3 & 2 / 3
\end{array}\right]\)
Exercise \(\PageIndex{49}\)
Find the column player's optimal strategy.
- Answer
-
\(\left[\begin{array}{l}
1 / 6 \\
5 / 6
\end{array}\right]\)
Exercise \(\PageIndex{50}\)
Find the value of the game if the row and column players' strategies are \([.7 \quad .3]\), and \(\left[\begin{array}{l}
.5 \\
.5
\end{array}\right]\), respectively.
- Answer
-
-.35