1324 Exam 4 Review. C(x) = x

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1 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, Exam 4 Review 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. Healthy Happy People Society just marketed GORUN tracker that keep track of calories burned, steps taken, etc... Researchers have found that the price-demand and cost function for these GORUN trackers are as follows: p(x) =75 2.5x, C(x) = x where x is in thousands of trackers and C(x) is in thousands of dollars. (a) How many GORUN trackers will it take to maximize the revenue? (b) What is the maximum revenue? (to nearest dollar) (c) What is the break-even point(s)? (d) What would the profit be if 9000 GORUN trackers are produced and sold? (to nearest dollar) (e) What should the GORUN trackers be priced at in order to maximize the profit? (round to nearest cent) 2. Find the domain for the following functions. Write you answer in interval notation. (a) y = 4x 8x 5 (b) f(x) = x (c) g(x) = 3 3x +4 (d) h(x) =5x 8 4x 2 (e) y =log 5 (8 6x) (f) f(x) =ln(2x 28) 3x +2 (g) y = x 4 8 (h) f(x) = 5x 2 33x 14 (i) y = 49 x 2 3. The revenue and cost functions of producing units are given by R(x) = 2x 2 +64xand C(x) =4x Determine the following. a) Minimum break-even quantity. b) Maximum revenue. c) Maximum profit. d) Number of units to maximize profit. 4. A = [ B= C= [ 2-3 h 4 p 6 Using the above matrices, find the value of h and p if CB = A.

2 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, Billy Bob Jenkins has been saving his coins. he has a total of 190 coins consisting of nickels, dimes, and quarters. He has four times as many nickels as he does quarters. If he has $17.85 altogether, how many nickels does he have? 6. Matrix A below shows the number of milkshakes sold at a local ice cream parlor during a particular week. The small milkshakes sell for $2.75, the medium sell for $3.25, and the large sell for $4. Find a matrix, P, that represents the prices and use it to find the matrix product for the revenue. How much revenue did the ice cream parlor make that week on chocolate milkshakes? Chocolate Small 40 Medium 30 Large 20 A= Vanilla Strawberry The profit function for a hot dog vendor can be given by P (x) = x2 2 the following. a) Maximum profit. b) Number of hot dogs necessary to maximize profit. +35x Determine 8. Bob operates three self-service gas stations in different parts of town. On a particular day, station A sold 600 gal of premium, 800 gal of super, 1000 gal of regular gasoline, and 700 gal of diesel fuel; station B sold 700 gal of premium, 600 gal of super, 1200 gal of regular gasoline, and 400 gal of diesel fuel; station C sold 900 gal of premium, 700 gal of super, 1400 gal of regular gasoline, and 800 gal of diesel fuel. This data can be represented by the following matrix. S = P S R D where the Rows are Row 1 = A, Row 2 = B, Row 3 = C Assume that the price of gasoline was $1.60/gal for premium, $1.40/gal for super, and $1.20/gal for regular, and diesel fuel sold for $1.50/gal. (adapted from #41, pg. 165 Finite Mathematics by Tan, edition 8) (a) Write the matrix, T, representing the unit prices for the various fuels. (b) Use these matrices to compute the total revenue matrix. What was the total revenue for Station C? 9. Given f(x) =3x 2 +6x 8, find (a) y-intercept (b) x-intercept(s) (c) vertex (d) axis of symmetry 10. Solve the following for x: (a) 27 x = (b) 2 6 x =4 2+x 11. Rewrite each log equation in exponential form. (a) log b x =20 (b) ln P =5

3 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, (c) log K = m 12. Rewrite each exponential equation in logarithmic form. (a) e 5 = J (b) H =5 a (c) y =10 x 13. The marketing research department for a company that manufactures and sells notebook computers established the following price-demand function: p = x where p is the wholesale price in dollars at which x thousand computers can be sold. (round all answers to 2 decimal places) (a) Find the Revenue function. (b) Find the output that will produce the maximum revenue. (c) What is the maximum revenue? (d) What is the wholesale price per computer (to the nearest dollar) that produces the maximum revenue? 14. Find the domain for f(x) = log 5 (3x 2) in interval notation. 15. Solve for x EXACTLY (no decimals). (a) ln(x + 14) 5=3 (b) 4e x+5 =16 (c) 3 4 x =21 (d) log 2 (x +8)=5 (e) x =5 16. The supply function for a product is given by 2x 9p + 40 = 0 and the demand function is given by x +4p 68 = 0. What is the equilibrium price for this product? 17. The marketing research department for a company that manufactures and sells memory chips for microcomputers established the following Revenue and Cost functions: R(x) =x(75 3x) C(x) = x where x is in millions of chips, and R(x) andc(x) are in millions of dollars. (a) Find the break even point(s). (b) What is the maximum revenue? (c) What is the Profit Function? (d) What is the maximum Profit and where does it occur? 18. Alex decides to set up a drink stand outside the Academic Building where he sells bottles of water, iced tea, and Gatorade. He sells each bottle of water for $1.25, each bottle of iced tea for $1.75, and each bottle of Gatorade for $2.50. At the end of the first day, he has revenue of $248 from the sale of 152 total bottles. He knows that he sold three times as many bottles of water as Gatorade. How many bottles of water and iced tea did he sell combined?

4 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, The quantity demanded of a certain brand of DVD player is 3000/wk when the unit price is $485. For each decrease in unit price of $20 below $485, the quantity demanded increases by 250 units. The suppliers will not market any DVD players if the unit price is $300 or lower. But at a unit price of $525, they are willing to make available 2500 units in the market. The supply and demand equations are known to be linear. (a) Find the demand equation. (b) Find the supply equation. (c) Find the equilibrium quantity. (d) Find the equilibrium price. 20. Find the domain for h(x) = 4x 2 in interval notation. x Solve the following for x: (a) 27 = 9 2x (b) 8 = 4 x2 2 5x 22. A running store sells brands of GU (packaged energy for runners), each in three flavors (chocolate, vanilla, and berry). Last weekend the store kept track of how many GU s they sold of each type. For Brand I, the store sold 15 chocolate, 10 vanilla, and 30 berry. For Brand II, the store sold 35 chocolate, 20 vanilla, and 40 berry. For Brand III, the store sold 35 chocolate, 50 vanilla, and 25 berry. On average, the chocolate, vanilla, and berry GU s sell for $2, $2.20, and $2.40 respectively. (a) Write the matrix, G, representing the number of GU s sold last weekend. (b) Write the matrix, P, representing the average prices for the various GU. (c) Use these matrices to compute the total revenue matrix. What was the total revenue for Brand I of the GU? 23. Bobby and Tony decide to manufacture pillows. The fixed costs for this pillow escapade is $4,000. It will cost them $10 to produce each pillow, but they plan to sell them at $30 each. (a) If they manufacture and sell exactly 327 pillows, how much of a profit/loss could they expect? (b) How many pillows must Bobby and Tony sell in order to break even? x Find the domain for f(x) = in interval notation. x A factory produces road bikes that each require 3 labor-hours in the assembly department, 1 labor-hour in the finishing department, and 2 labor-hours in the shipping department. This factory also produces mountain bikes that each require 5 labor-hours in the assembly department, 1.5 labor-hours in the finishing department, and 1 labor-hour in the shipping department. This factory also produces trick bikes that require 1 labor-hour in the assembly department, 3.5 labor-hours in the finishing department, and 2 labor-hours in the shipping department. If our rows represent Road Bike, Mountain Bike, and Trick Bike (row 1, 2, 3 respectively) and our columns represent assembly, finishing, and shipping (columns 1, 2, 3, respectively), then we have L =

5 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, Assembly personnel receive $9/hour, finishing personnel receive $6/hour, and shipping personnel receive $5/hour. (a) Set up the matrix, P, representing the pay of the personnel. (b) Use matrices L and P to find the total labor cost for assembling a road bike. [ 4 8 [ 5 k Let A = B = C = m -2 5 x Given that AB = C, find the value for m. 27. Solve for x: 4+log 4 (3x +1)=7 28. Let p(x) = x and C(x) =2x be the price-demand equation and the cost function, respectively for the manufacturing of x umbrellas. (a) What is the Revenue Function? (b) What are the break even point(s)? (c) What is the Profit Function? (d) What is the maximum Profit? (e) How many umbrellas are needed to maximize the profit? (f) What should the umbrellas be priced at to reach the maximum profit? 29. Solve 17(2) 3x =8 30. Given f(x) = 4x 2 24x 20, find (a) y-intercept (b) x-intercept(s) (c) vertex (d) axis of symmetry 31. Find the domain of: (a) g(x) =e 2x 5 (b) f(x) = 3x 2 x Solve and write your answer in (x,y,z) form. If there is a unique solution, what is it? If there is no solution, write no solution. If there are an infinite number of solutions, use parametric form for your answers and give an example of ONE specific solution. 6x - 4y + 10z = y + 5z 2 = Solve ln(5x) =2 [ 5 a 34. Let E = b 3 F= [ c -1 and G = Given that EF=G, find the values of a, b, and c. 35. Solve 10 4x =6. [ Kate, Chelsea, and Anmarie went Trick-or-Treating together. Anmarie got twice as much candy as Chelsea. The amount of candy Kate got was half of what Chelsea and Anmarie got combined. Altogether they received 648 pieces of candy. How many pieces of candy did Kate get?

6 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, In deciding whether to set up a new manufacturing plant, company analysts have decided that a linear function is a reasonable estimation for the total cost, C(x), in dollars to produce x items. They estimate the cost to produce 10,000 items as $547,500, and the cost to produce 50,000 items as $737,500. (a) Find a formula for C(x). (b) Find the total cost to produce 100,000 items. (c) How many items are produced if the total costs are $502,289.50? (d) Find the marginal cost of the items to be produced in this plant and what does it mean to the manager? 38. Find the domain of ln(4x 9) in interval notation. 39. Find the equation of the line going through the points (350,920) and (410,770). 40. The price-demand function and cost function for a certain product are given below p(x) =60 1.5x, C(x) =9x where x is in thousands of units, p(x) is in dollars, and C(x) is in thousands of dollars. (a) What is the maximum revenue? (to nearest dollar) (b) At what price will the Revenue be maximized? (c) What is the break-even point(s)? (d) How many units does it take to maximize the profit? (e) What is the profit when 15,000 units are produced/sold? 41. The quantity demanded for a certain brand of washing machine is given by p = 3 8 x and supply equation for the washing machine is given by p = 2 3 x + 400,wherex is the quantity 3 demanded and p is the price. What is the equilibrium price for the washing machine? ( ) x Solve 2 x2 4x = Fred and George produce and sell Wacky Wands. They discovered the price-demand equation to be p(x) = 0.05x +45 where p(x) is the price of the wands (in dollars) and x is the number of wands. The cost equations for wacky wands is given by C(x) = x, wherec(x) is the cost of the wands (in dollars) and x is the number of wands. (a) How many wands must they sell in order to maximize the revenue? (b) How many wands must Fred & George produce & sell to break even? (c) What is the maximum profit they could make on these wands? (d) What should the price of the wands be in order for them to maximize profit?

7 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, When multiplying AB for the matrices below, which element is in the C 12 spot of the product matrix, C? A = [ 4-3 2p k 6 -w B = A company with manufacturing plants in California and Texas has labor-hour and wage requirements for the manufacture of two inexpensive calculators as given in matrices M and N below: Labor Hour Per Calculator Hourly Wages Fabricating Department Assembly Department Packaging Department California plant Texas plant M= 0.15 hr 0.10 hr 0.05 hr Model A 0.25 hr 0.20 hr 0.05 hr Model B N= $12 $10 $15 $12 Fabricating Assembly $7 $6 Packaging Decide how to multiply the matrices above in order to find the total labor cost for each model calculator in each state, then find the labor cost for producing one model B calculator at the California plant. 46. Find the domain of f(x) = 47. Solve (e 3 ) 2x = e x+5 5 3x 2 +2x 1 in interval notation. 48. A company markets potato launchers. They have discovered that the demand for these launchers is given by p = x. The cost function for the company is given by C(x) =63x +42, 500. Hint: x-max=1500, y-max=200,000 (a) What is the revenue function? (simplify) (b) How many potato launchers would it take for the company to break-even? (round to nearest whole number) (c) What should the potato launchers be priced at in order to maximize revenue? (round to nearest dollar) (d) Find the profit equation (and simplify). (e) What is the maximum profit? (round to nearest cent...aka 2 decimal places) [ k Let A = and B = m j -3 When you multiply AB, which element is in the 1st row, 2nd column of the product matrix?

8 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, Solve log 4 (5x +1)=2 51. Given f(x) =6x 2 +96x + 60, find (a) y-intercept (b) x-intercept(s) (c) vertex (d) axis of symmetry 52. If 20 lb of rice and 10 lb of potatoes costs a total of $13.50, and 30 lb of rice and 12 lb of potatoes costs a total of $18.78, how much will 24 lb of rice and 50 lbs of potatoes cost? 53. The matrix below is in RREF. What would the solution be for this system of linear equations? If none, write none. If infinite, write in parametric form and give at least 2 examples. x y z w Suppose that the demand and price for strawberries are related by p = D(q) =5 0.25q where p is the price (in dollars) and q is the quantity demanded (in hundreds of quarts). (a) Find the price when the demand level is 840 quarts. (b) Find the quantity demanded for the strawberries when the price is $2.40. Suppose the price and supply of strawberries are related by p = S(q) =0.25q where p is the price (in dollars) and q is the quantity demanded (in hundreds of quarts) of strawberries. (c) Find the quantity supplied at a price of $2. (d) What is the equilibrium quantity for the strawberries? (e) What is the equilibrium price for the strawberries? 55. The matrices below are all in RREF. Write out the solution(s) for each matrix. If none, write none. If infinitely many, write out the parametric solution. x y z x y z w (a) (b) Solve 2 x+1 =6 x A= -4 m -2 6 B= k Find the product of BA using matrices above. Which element is in: (a) Row 3, Column 1 (b) Row 1, Column 2 (c) Row 3, Column Solve without a calculator. Write answers EXACTLY (no decimals).

9 c Dr. Patrice Poage and Mrs. Reanna Carr, June 26, (a) 5 x 8 =5 3 (b) 2 x+5 =64 (c) 3 x2 8 =9 x 59. Dr. Poage has decided to produce and sell rings that say 1 MATH! The price-demand equation and cost function for the production of these rings are given respectively by: p = x , C(x) =72, x where x is the number of rings that can be sold at a price of $p per ring and C(x) is the total cost (in dollars) of producing x rings. (a) Determine the break even point(s). (b) What is the maximum Revenue and how many rings are needed to maximize the Revenue? (c) What should the rings be priced at in order to maximize the Revenue? (d) How many rings are needed to maximize the Profit and what is the maximum Profit? (e) What should the rings be priced at in order to maximize the Profit? 60. Mason has been saving his money. He has half as many nickels as pennies and quarters combined. He has four times as many pennies as dimes. If he has a total of 108 coins and the total face value of these coins is $7.28, how many nickels and dimes does he have combined?