Math 1070 Sample Exam 1 Spring 2015

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University of Connecticut Department of Mathematics Spring 2015 Name: Discussion Section: Read This First! Read the questions and any instructions carefully.

1 University of Connecticut Department of Mathematics Spring 2015 Name: Discussion Section: Read This First! Read the questions and any instructions carefully. The available points for each problem are given in brackets. You must show your work to obtain full credit (and to possibly receive partial credit). Calculators are allowed, but you must still show your work. Make sure your answers are clearly indicated, and cross out any work you do not want graded. If you finish early, check all your solutions before turning in your exam. Grading - For Administrative Use Only Page: Total Points: Score:

2 1. A bakery selling cupcakes has the cost equation C = 0.9x + 620, where x is the number of cupcakes. It is known that the bakery makes a profit of $250 from selling 300 cupcakes. Assume linear cost and revenue models. (a) Find the revenue equation. [6] (b) Find the break-even quantity. 2. A clothing store finds that there is demand for 80 sweaters when they are priced at $31 each, and the demand increases to 100 when they are priced at $27 each. Assume linear supply and demand models. (a) Find the demand equation. [5] Page 1 of 7

3 (b) If the supply equation is p = 0.3x + 16, find the equilibrium quantity and price. [4] 3. In a survey of 200 people, it was found that 150 own a car, 80 own a bike, 30 own rollerblades, 45 own a car and a bike, 20 own a bike and rollerblades, 10 own a car and rollerblades, and 40 own both a car and a bike but not rollerblades. (a) Represent all of this information in a complete Venn diagram. [6] (b) If someone is randomly selected from those surveyed, what is the probability that they [4] own exactly two of the three? (c) If someone is randomly selected from those surveyed, what is the probability that they [4] own rollerblades or a bike (or both) but not a car? Page 2 of 7

4 4. A diner makes beef burgers and turkey burgers. Each beef burger requires 3 minutes of cooking time and earns $5 in revenue. Each turkey burger requires 5 minutes of cooking time and earns $6 in revenue. The diner can make at most 100 burgers total, and has 450 minutes of cooking time available. How many of each type of burger should be made and sold to maximize revenue? (a) Describe this as a linear programming problem by defining appropriate variables, and [5] giving the objective function and inequalities. (b) Sketch the feasible region and find its corner points. [8] Page 3 of 7

5 (c) Determine how many of each type of burger should be made and sold to maximize revenue. [4] 5. A subcommittee of 6 people must be formed from a committee of 8 women and 7 men. (a) How many different subcommittees are possible? [2] (b) How many different subcommittees are possible with exactly 3 women? (c) How many different subcommittees are possible with at least 2 women and at least 2 men? [5] Page 4 of 7

6 6. Let U = {a, b, c, d, e, f, g, h, i, j}, A = {a, b, e, f, i, j}, B = {d, e, f, g}, and C = {b, d, f, h}. Find each of the following sets. (a) (A B) c (b) (A C) B c 7. Suppose we draw a five-card hand from a standard 52-card deck. (a) How many ways are there to get a full house (a three of a kind and a two of a kind, eg. [4] three Aces and two 5s)? (b) How many ways are there to get a three of a kind, and two cards of different values that [5] are different from the three of a kind (eg. three 5s, one Queen, one 2)? Page 5 of 7

7 8. Let E and F be two events, and suppose P (E F ) = 0.7, P (E) = 0.4, and P (F ) = 0.5. (a) Represent this information in a complete Venn diagram (b) Find P (E F c ) [2] (c) Find P (E c F ) [4] 9. A family consisting of two parents and five children are going to line up in a row to take a family photo. (a) How many ways are there for the seven family members to line up? [2] (b) How many ways are there for the seven family members to line up if the two parents must [4] be next to each other? Page 6 of 7

8 10. We toss a tetrahedral (four-sided) die twice, and observe the number showing on each toss. (a) Write out all elements in the sample space. [4] (b) What is the probability that the sum of the two observed values is 4? (c) What is the probability that the sum of the two observed values is at least 4? [4] Page 7 of 7

Math 1070 Sample Exam 1

University of Connecticut Department of Mathematics Math 1070 Sample Exam 1 Exam 1 will cover sections 1.1, 1.2, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 4.4, 4.5, 5.1 and 5.2. This sample exam is intended to be