Math 111 Final Exam, Autumn 2013 HONOR STATEMENT

Transcription

1 NAME: QUIZ Section: STUDENT ID: Math 111 Final Exam, Autumn 2013 HONOR STATEMENT I affirm that my work upholds the highest standards of honesty and academic integrity at the University of Washington, and that I have neither given nor received any unauthorized assistance on this exam. SIGNATURE: Do not open the test until instructed to do so. Please turn your cell phone OFF now. This exam consists of this cover sheet followed by eight problems on nine pages. When the test starts, check that you have a complete exam. This exam is closed book. You may use one double-sided, handwritten page of notes, a ruler, and a calculator. Put everything else away. You may not share notes. Unless otherwise indicated, you must show your work and justify your answers. The correct answer with incomplete or missing supporting work may result in no credit. If you use a guess-and-check method when a better method is available, you may not receive full credit. On graph-based problems, show your work clearly by marking all lines and points that you use. Place your final answer in the indicated spaces. Unless otherwise specified, you may round your final answer to two digits after the decimal. Do NOT round off before your final answer. If you need more room, use the backs of the pages and indicate to the grader that you have done so. Problem Total Points Score Total 100

2 1. (10 points) The following graph shows the distance, in miles, traveled by a Truck at t minutes. Draw and label all the lines you use on the graph provided. Recall that 1 hour is 60 minutes. (a) Estimate the Average Speed of the Truck during the last 90 minutes of its journey. miles per minute (b) Find a 2 hour time interval when the Average Speed of the truck is 1 mile per minute. From t = to t = minutes (c) Find the Average Speed of the Truck during the 1 minute interval at the end of the 2nd hour of its journey. miles per minute (d) If a Car starts at the same place as the Truck and goes at a constant speed of 1.25 miles per minute, when will it catch the Truck? After (e) Estimate the greatest distance between the Car in part (d) and the Truck. minutes miles

3 2. (12 pts) Below is the graph of the Total Cost for producing q Pragmas. Label the lines you draw. Note that the Total Cost is in THOUSANDS of dollars. Include units in your answers. (a) (3 pts) Estimate the Average Cost at 150 Pragmas. (b) (3 pts) Estimate the Breakeven Price. (c) (2 pts) Estimate the Variable Cost of producing 550 Pragmas. (d) (4 pts) At approximately how many Pragmas is the Average Variable Cost (AVC) equal to 30 dollars per Pragma?

4 3. (14 points) The Average Variable Cost and Marginal Cost (in dollars per item) for producing q Whatnots are given by the following formulas: AV C(q) = 0.004q 2 3q MC(q) = 0.012q 2 6q (a) (3 pts) If the Total Cost of producing 400 Whatnots is $96,000, what is your Fixed Cost? (b) (3 pts) What is the Shutdown Price? $ dollars per Whatnot (c) (2 pts) What is the value of T C(351) T C(350)? (d) (6 pts) If you sell each Whatnot for $270, what is your maximum profit? $

5 4. (10 points) Do not round answers in the following questions. Use fractions when necessary. (a) Solve for x the following equation: 7 3 x 2 = 6 3x x = (b) Find the equation of the line through the two points (2, 3) and ( 5, 12). Give your answer in the form y = mx + b. What do m and b stand for? y = m is the b is the, and

6 5. (16 points) White Powder Products makes downhill and cross-country skis. A pair of downhill skis requires 2 man-hours for cutting, 1 man-hour for shaping and 3 man-hours for finishing. A pair of cross-country skis requires 2 man-hours for cutting, 2 man-hours for shaping and 1 man-hour for finishing. Each day the company has available at most 140 man-hours for cutting, 120 manhours for shaping and 150 man-hours for finishing. How many pairs of each type of ski should the company manufacture each day in order to maximize profit if a pair of downhill skis yields a profit of $10 and a pair of cross-country skis yields a profit of $8? (a) (1 pt) Define the variables: x = y = (b) (2 pts) Give the formula for the objective function: f(x, y) = (c) (3 pts) Write down the constraint inequalities. (d) (4 pts) Graph the constraints and clearly shade the feasible region. (this problem continues on the next page)

7 (continued from the previous page) (e) (4 pts) Compute the coordinates of the all the vertices of the feasible region. Show your work! ANSWER (list all): (x, y) = (f) (2 pts) How many pairs of each type of ski should the company manufacture each day in order to maximize profit? Show all work. pairs of downhill skis, and pairs of cross-country skis

8 6. (12 points) (a) Compute the sum of the first 200 terms of the sequence A: 4, 6, 8, 10,... Sum = (b) Compute the 9th term of the sequence B: 729, 243, B 9 = (c) A city doubles its population every 100 years. How long does it take this city to triple its population? years (d) After knee surgery, your trainer tells you to return to your jogging program slowly. She suggests jogging for 12 minutes each day during the first week, and each week thereafter to increase your daily jogging time by 8%. After 10 weeks of following this schedule, and assuming your average jogging speed is 0.2 miles/minute, what distance will you cover on your daily jog? miles

9 7. (14 points = 4 pts + 4 pts +6 pts) (a) You have $1200 to deposit into an account in which interest is compounded continuously. What is the smallest annual interest rate the account can have if you are to have $1800 in the account after 7 years? r 100% = % (b) Peter Parker deposited $1000 in an account that earned simple interest at an annual rate of 5% and left it there for 5 years. At the end of the 5 years, Peter deposited the entire amount from that account into a new account that earned 3.6% compounded quarterly. He left the money in this account for 6 years. What is Peter s balance after the 11 years? $ (c) On Jan. 1, Tony Stark has a balance of $6, on a credit card that charges an annual rate of 18%, compounded monthly. His minimum monthly payment is $125. If he pays $125 at the beginning of each month, starting in January (and makes no other charges or payments), how long does it take him to pay off this card? How much interest does he end up paying? months; The interest paid is $

10 8. (12 points) Selina Kyle just turned 40 years old and plans to retire at 65. She has a savings account that pays 10% compounded annually. She wants to figure out how much she needs to put aside in this account at the end of each year for the next 20 years, so that she will be able to withdraw $50,000 per year for 20 years after she retires, with the first withdrawal on her 66th birthday. (a) Compute the lowest amount Selina must have in her account when she turns 65, in order to be able to withdraw $50,000 at the end of each following year for twenty years. Recall that her account pays 10% compounded annually. $ (b) Between 60 and 65 years, Selina is planning to make no deposits and no withdrawals from this same account. What balance should she have in the account at 60 years old? $ (c) How much does she need to deposit at the end of each year in this account for twenty years, starting at 40, to reach the necessary balance by the time she s 60? $