1324 Exam 2 Review Covers Chapter 5 and Chapter 4

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1 c Dr. Patrice Poage, August 30, Exam 2 Review Covers Chapter 5 and Chapter 4 NOTE: This review in and of itself does NOT prepare you for the test. You should be doing this review in addition to studying all your suggested homework, quizzes, and OLHW. 1. Upon graduating from college, Cooper finds he owes $56,000 in student loans with an interest rate of 4.8% compounded monthly. He must pay the loans off in 10 years. How much interest will he end up paying for this loan? 2. Kevin and Leslie buy a house for $162,500. They have no money to put as a down payment, so they finance the whole thing at 7.3%/year interest rate compounded monthly for 30 years. (a) What are their monthly payments? (b) Suppose they bought this house seven years ago. What is their current outstanding principle? (aka, how much money do they still owe?) 3. Jennie and Abigail own a local beauty parlor that makes two styles of hair dryers, the Petite and the Deluxe. It requires 1 hour of labor to make the Petite and 2 hours of labor to make the Deluxe. The materials cost $4 for the Petite and $3 for the Deluxe. The profit is $3 for the Petite and $8 for the Deluxe. The company has 4100 labor-hours available each week and a materials budget of $10,350 per week. How many of each dryer should be made each week to maximize the profit? This problem can be set up in the following manner: x = the number of Petite dryers y = the number of Deluxe dryers Maximize: P=3x+8y x + 2y x + 3y x 0, y 0 (a) Set up the initial Simplex Tableau for this problem. (b) Circle the pivot element, pivot, write down the new tableau. (c) Repeat the previous step until you reach your final tableau. (d) Answer the following questions based upon your final tableau: What is the maximum profit? How many Deluxe dryers are needed to maximize the profit? How many Petite dryers are needed to maximize the profit? Are there any resources leftover? If so, what and how much? 4. Taryn deposits $2400 into a bank account paying simple interest. Three months later she has $2435 in the account. Assuming no other deposits or withdrawals, what is the interest rate on the account? (write as a percentage with 2 decimal places)

2 c Dr. Patrice Poage, August 30, You want to buy a new car in 5 years so you decide to start a sinking fund that is compounded weekly at 2.8%. If you want to have $15,000 in five years, (a) How much money should you put into the account each week? (b) How much interest will you end up earning on the account? 6. The following MAXIMIZATION Linear program problem is in final form. Write out solutions. x y z u v w P C Nine years ago, Connor and Jennifer bought a house for $275,000. They paid 30% down and financed the remaining balance at 3.75% annual interest rate compounded monthly for 30 years. (a) What are their monthly payments? (b) What is their current outstanding principle? (c) If they make the minimum monthly payment each month, how much interest will they end up paying on this house? (d) If they decided from the beginning that they wanted to pay the house off in 20 years instead of 30, how much should their monthly payments be? (e) How much money would they end up saving by paying their house off in 20 years instead of 30 years? 8. Freeeman Florists makes two kinds of special Christmas flower arrangements: Peppermint Bouquet and Angel Bouquet. Each Peppermint Bouquet requires 1 Rose, 3 Lillies, and 4 Carnations. Each Angel Bouquet requires 2 Roses, 2 Lillies, and 1 Carnation. Freeman Florists has 24 Roses, 32 Lillies, and 36 Carnations in stock today. The Peppermint Bouquets sell for $27/each while the Angel Bouquet sells for $15/each. Assuming all bouquets made will sell, how many of each type of bouquet should they make today in order to maximize today s revenue. Let x = # of Peppermint Bouquets Let y = # of Angel Bouquets Maximize R = 27x + 15y x + 2y 24 3x + 2y 32 4x + y 36 x 0 y 0 (a) Set up the initial Simplex tableau. (b) Circle pivot elements and show each new tableau. (c) What is the maximum revenue? (d) How many Peppermint Bouquets are needed to maximize the revenue? (e) How many Angel Bouquets are needed to maximize the revenue? (f) Are there any resources leftover? If so what and how much?

3 c Dr. Patrice Poage, August 30, Cheryl has a credit card debt of $4,900. Interest is being accrued at 18.9% compounded monthly on the unpaid balance. Cheryl cuts up the card (thus making no more charges on it) and starts paying the minimum monthly payment of $79/month. (a) If she only pays the minimum payment each month, how many years will it take her to pay off the credit card? (b) How much interest will she end up paying on this credit card? (c) After 6 months of payments, how much has she paid off towards the balance (debt) on her credit card? 10. Upon graduation (age 18), Rory s parents gave her $8000 which she invested in an account that is compounded weekly. At age 21, she has $9634 (without having made any more deposits or withdrawals). At what interest rate was this money invested? (write as a percentage with 1 decimal place) 11. Addison takes out a short-term loan for $11,500. It is compounded weekly at 8.3% interest. He wants to pay it off using equal payments over the course of 5 years. How much interest will he end up paying on this loan? 12. The Hammond family got a tax refund of $2300 this year and decided now was a good time to start a sinking fund so they can have $20,000 in 4 years and put in a swimming pool. They start an account, using the tax refund as the initial deposit. The account earns 1.4% interest compounded monthly. How much money should they put into the account, each month, in order to reach their goal? 13. Reese deposits $1500 into an account that pays interest of 2% per year compounded quarterly. She doesn t touch the money for 3 years. Then she moves all her money to a different account that pays 3% interest compounded quarterly and keeps it there 5 years. How much money does she have at the end of all this? 14. Sophie deposits $7000 into an account earning simple interest. What is the interest rate if 7 months later she has $ in the account. 15. Using the simplex tableau below, perform the next pivot operation and determine what number is in the 3rd row / 2nd column of your new tableau. x y z u v P C Patrice notes that the price of a Rt 44 Sonic drink seems to increase each year by about 1.5%. If the price is currently $2.37, and keeps increasing at this rate, how much will these Rt 44 Sonic drinks be in 4 years? 17. Andra figures she can afford a car payment of $320/month if the interest rates remain at 1.9%/year compounded monthly. If she plans to put $3200 down and finance the rest for 5 years, what is maximum price cars she should even consider?

4 c Dr. Patrice Poage, August 30, Bayer Pharmaceutical produces three kinds of cold formulas: I, II, and III. It takes 2.5 hr to produce 1000 bottles of formula I, 3 hr to produce 1000 bottles of formula II, and 4 hr to produce 1000 bottles of formula III. The profits for each 1000 bottles of Formula I, II, and II are $180, $200, and $300 respectively. Suppose for a certain production run, there are enough ingredients on hand to make at most 9000 bottles of formula I, 12,000 bottles of formula II, and 6000 bottles of formula III. Furthermore, suppose the time for the production run is limited to a maximum of 70 hrs. How many bottles of each formula should be produced in this production run so that the profit is maximized? Maximize P = 180x + 200y + 300z 2.5x + 3y + 4x 70 x 9 y 12 z 6 x 0, y 0, z 0 (a) Set up your Initial Tableau. (there are 4 slack variables) (b) Use simplex method (Show intermediate steps and circle ALL pivot elements along the way. Be prepared to pivot 3, 4, or 5 times) (c) Answer the questions at bottom of page. NOTE: Don t forget to convert the number of bottles of formula back into thousands. The maximum profit is when bottles of formula I, bottles of formula II, and bottles of formula III are made. Are there any resources leftover? If so what, and how much? 19. Suppose a population gathers plants and animals for survival. They need at least 420 units of energy, 350 units of protein, and 7 hides during some time period. One unit of plants provides 30 units of energy, 10 units of protein, and no hides. One animal provides 20 units of energy, 25 units of protein, and 1 hide. Only 28 units of plants and 28 animals are available. It costs the population 34 hours of labor to gather one unit of plant and 21 hours for an animal. How many units of plants and how many animals should be gathered to meet the requirements and yet minimize the number of labor hours. DEFINE YOUR VARIABLES and JUST SET UP the Linear Programming problem. DO NOT SOLVE!! 20. Four years ago, Mason bought a boat for $74,500. He financed the whole thing at 4.25% annual interest rate compounded monthly for 15 years. What is the current outstanding principle on the boat? 21. Dillon and Katie bought a house for $286,000. They paid 15% down and financed the rest at 3.9% interest compounded monthly for 30 years. (a) What are their monthly payments? (b) How much of their very first payment went towards the debt of the loan?

5 c Dr. Patrice Poage, August 30, (c) How much of their very first payment went towards the interest of the loan? (d) How much interest did they end up paying on this house? (e) If, from the start, they paid an extra $150 towards their payment each month, how long many years would it take them to pay off the house? Round to ONE decimal. (f) How much money would they save by doing this (paying extra $150/month towards payment)?

6 c Dr. Patrice Poage, August 30, Ryan and ReAnna bought a house. They put a 30% down payment on the house that cost $185,000, and financed the rest at 5.7% interest compounded monthly for 30 years. (a) What are their monthly payments? (b) How much of their very first payment went towards the principle of the loan? (c) How much of their very first payment went towards the interest of the loan? (d) If they paid $100 more then the minimum monthly payment, how many years will it take to pay off the loan? 23. Use the following linear programming problem to answer the following questions. Maximize P = 3x + 2y 8z 2x + 7y z 38 x 4y 60 3x + 2y + 5z 55 x 3y + z 73 x 0, y 0, z 0 (a) How many slack variables would the following linear programming problem have? (b) Write the initial tableau for this problem. (c) Circle the first pivot element. 24. The DiMarco family puts a lump sum into an account that pays 8.3% interest compounded semi-annually. How long will it take to triple this amount? (round to one decimal place) 25. Willie invests $6000 into an account earning 7% per year simple interest. How many years will it take for the future value to become $7134? (round to one decimal place) 26. Twila wants to put money in a savings account now so that she will have $1800 in 5 years. The savings bank pays 6% interest compounded quarterly. How much should she invest? 27. On her 22nd birthday, Marisa deposits $5,000 in an account that pays 5.6% compounded daily. How much will be in her account on her 40th birthday is she makes no more deposits or withdrawals? 28. At age 15 Tanner decides to start putting $20 into a savings account each week. The account has an interest rate of 3.9% compounded weekly. (a) How many years will it take him to accumulate $10,000 in this account? (round to 1 decimal place) (b) How much interest will him have earned over this time? 29. Jack and Rose are putting in a swimming pool. They are able to make a down payment of $8000 and finance the remaining balance with a loan at 6.3% per year compounded monthly. Under the terms of the finance agreement, they are required to make payments of $495/month for the next 4 years. (a) What was the cash price for putting in this swimming pool? (b) How much interest will the end up paying?

7 c Dr. Patrice Poage, August 30, A company manufactures outdoor furniture consisting of regular chairs, rocking chairs, and chaise lounges. Each piece of furniture passes through three different production departments: fabrication, assembly, and finishing. Each regular chair takes 1 hour to fabricate, 2 hours to assemble and 3 hours to finish. Each rocking chair takes 2 hours to fabricate, 2 hours to assemble, and 3 hours to finish. Each chaise lounge takes 3 hours to fabricate, 4 hours to assemble, and 2 hours to finish. There are 2800 labor-hours available in the fabrication department, 3200 in the assembly department, and 3500 in the finishing department. The company makes a profit of $25 on each regular chair, $24 on each rocking chair, and $31 on each chaise lounge. How many chairs of each type should the company produce in order to maximize profit? NOTE: the problem has been set up for you as follows Maximize P = 25x + 24y + 31z x + 2y + 3z x + 2y + 4z x + 3y + 2z 3500 x 0, y 0, z 0 (a) Set up your Initial Tableau and circle 1st pivot element. (b) Use simplex method (show intermediate steps, circling each pivot element) (c) Answer the questions at bottom of page. Answers: The maximum profit is, when they make regular chairs, rocking chairs, and chaise lounges Are there any resources leftover? If so WHAT, and HOW MUCH? 31. Taylor decides to start an annuity. She deposits $15/week into an account that is compounded weekly at 1.5% interest rate. How much interest would she earn on this annuity if he continues this for 15 years? 32. McKenna bought a house for $205,000. She paid 10% down and financed the rest at 3.85% interest compounded monthly for 30 years. How much of her very first payment went towards the principle of the loan? 33. Using the simplex tableau below, which element would be the next pivot element? x y z u v P C

8 c Dr. Patrice Poage, August 30, A pizza shop prides itself in making only two pizzas, a pepperoni pizza and a cheese pizza. Each pepperoni pizza (x) requires 1 cup of cheese and takes 6 minutes to make, while each cheese pizza (y) requires 2 cups of cheese and takes only 4 minutes to make. Each pepperoni pizza sells for $8 and each cheese pizza sells for $6. If 48 cups of cheese and 2 1/2 hours of time are available, how many of each type of pizza should the shop make in order to maximize the revenue? Write the objective function and restraints representing this linear programming problem. DO NOT SOLVE THE PROBLEM! 35. Alex wants to be able to give her parents $100 at the end of each quarter for the next 10 years. How much money should she deposit into an account today, at 2.75% interest compounded quarterly, to be able to make these payments to her parents? 36. Using the simplex tableau below, perform the next pivot operation and determine what number is in the 3rd row / 2nd column of your new tableau. x y z u v P C The following MAXIMIZATION problem is in final form. Write out the solutions IN THE BLANKS PROVIDED. x y z u v w P C MAX =, x =, y =, z =, u =, v =, w = 38. Shelbi has a credit card debt of $5,300. Interest is being accrued at 21.9% compounded monthly on the unpaid balance. Shelbi cuts up the card (thus making no more charges on it) and starts paying the minimum monthly payment of $98/month. Hint: Make the payment positive (a) If she only pays the minimum payment each month, how many years will it take her to pay off the credit card? (b) How much interest will she end up paying on this credit card? (c) After 6 months of payments, how much has she paid off towards the balance (debt) on her credit card? 39. Craig buys a new car for $15,750. He pays $3000 down and finances the rest at 5.25% compounded monthly. He has the choice to either spread his payments over 5 years or 6 years. How much money would he end up saving if he chose the 5 year plan versus the 6 year plan?

9 c Dr. Patrice Poage, August 30, Clearly circle the pivot element below, then pivot one time and write down the element in Row x y z u v w P C , Column 3 of your new tableau