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# Recent problems solved by 'Theo'

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 Finance/350135: Suppose you would like to be paid \$20,000 per year during your retirement, which starts in 20 years. Assuming the \$20,000 is an annual perpetuity and the discount rate is an effective 4% per year, how much should you save per year for the next 20 years so that you can achieve your retirement goal?1 solutions Answer 250267 by Theo(3458)   on 2010-09-30 10:54:18 (Show Source): You can put this solution on YOUR website!If you want to be paid \$20,000 per year for the rest of your life without the balance of the fund going down, and assuming that the interest rate at which the account will increase each year is 4%, and you expect to start withdrawing the money immediately in year 20, then the amount of money that needs to be in the fund is equal to \$520,000. That's equivalent to \$20,000 / .04 = \$500,000 plus \$20,000 to be withdrawn immediately to make a total of \$520,000 required to be in the account in year 20. If you have to figure in taxes, then the amount taken out each year will have to pay the going tax rate, and you will have to have more than \$500,000 in the account to start with in order to cover the withdrawal plus taxes on the withdrawal. We'll forget about taxes for sake of simplicity. You will need to have \$520,000 in the account 20 years from now. If this is year 0, then year 20 is an additional 20 years from now. If the investment rate is 4%, then you would need to invest \$237,321.212 today in order to have \$520,000 in the account in year 20. The formula to use to calculate the present value of the future amount of \$520,000 is shown below: PRESENT VALUE OF A FUTURE AMOUNT PV = Present Value FA = future amount i = Interest Rate per Time Period n = Number of Time Periods I used this formula to duplicate the result that I obtained through the use of a financial calculator. You should be able to duplicate the result also. i = .04 n = 20 FA = \$520,000 PV is calculated to be On a yearly cash flow analysis, this is what it would look like: ``` ACCOUNT ACCOUNT YEAR BALANCE WITHDRAWAL 0 \$237,321.21 \$0.00 1 \$246,814.06 \$0.00 2 \$256,686.62 \$0.00 3 \$266,954.09 \$0.00 4 \$277,632.25 \$0.00 5 \$288,737.54 \$0.00 6 \$300,287.04 \$0.00 7 \$312,298.52 \$0.00 8 \$324,790.47 \$0.00 9 \$337,782.08 \$0.00 10 \$351,293.37 \$0.00 11 \$365,345.10 \$0.00 12 \$379,958.91 \$0.00 13 \$395,157.26 \$0.00 14 \$410,963.55 \$0.00 15 \$427,402.10 \$0.00 16 \$444,498.18 \$0.00 17 \$462,278.11 \$0.00 18 \$480,769.23 \$0.00 19 \$500,000.00 \$0.00 20 \$520,000.00 \$20,000.00 21 \$520,000.00 \$20,000.00 22 \$520,000.00 \$20,000.00 23 \$520,000.00 \$20,000.00 24 \$520,000.00 \$20,000.00 25 \$520,000.00 \$20,000.00 ``` The money is invested in year 0 and grows at 4% per year until year 20 at which point 20,000 is withdrawn each year keeping the balance at a steady \$520,000. I went to year 25 to show you that the account balance remains the same, despite the \$20,000 withdrawal each year.
 Finance/350094: please help me! Suppose you expect to receive \$1,000 per year for each of the next 15 years, except that you will not receive any payments in years 3 and 5. What is the present value of this amount if the discount rate is 12% per year?1 solutions Answer 250247 by Theo(3458)   on 2010-09-30 10:14:55 (Show Source): You can put this solution on YOUR website!We'll assume you start from year 0. The present value of 15 years worth of payments would be \$6,810.864489. This includes payments for years 3 and 5. You need to subtract them. The present value for a \$1,000 payment in year 3 is equal to \$711.7802478. The present value for a \$1,000 payment in year 5 is equal to \$567.4268557. Subtract both these payments from the present valueof \$6,810.864489 and you have your answer. The answer would be \$5,531.657386 I used a financial calculator, but you'll get the same result by use of the present value of payments equation and the present value of a future value equation. For the 15 years of payments, use the present value of payments equation. For the present value of the missing payments, use the present value of a future value equation. A cash flow analysis, using the same technique, would show the following result: ```year payment present value of payment 0 1 \$1,000.00 \$892.86 2 \$1,000.00 \$797.19 3 \$1,000.00 \$711.78 4 \$1,000.00 \$635.52 5 \$1,000.00 \$567.43 6 \$1,000.00 \$506.63 7 \$1,000.00 \$452.35 8 \$1,000.00 \$403.88 9 \$1,000.00 \$360.61 10 \$1,000.00 \$321.97 11 \$1,000.00 \$287.48 12 \$1,000.00 \$256.68 13 \$1,000.00 \$229.17 14 \$1,000.00 \$204.62 15 \$1,000.00 \$182.70 sum of payments and present value of payments from year 1 through year 15 are shown below. \$15,000.00 \$6,810.86 missing payments and present value of apyments that need to be subtracted are shown below. 3 \$1,000.00 \$711.78 5 \$1,000.00 \$567.43 total minus missing payments and present value of missing payments is shown below. \$13,000.00 \$5,531.66 ``` the formulas you would use are shown below: PRESENT VALUE OF A PAYMENT PV = present value PMT = payment per time period i = interest rate per time period n = number of time periods PRESENT VALUE OF A FUTURE AMOUNT PV = Present Value FA = Future Amount i = Interest Rate per Time Period n = Number of Time Periods Using these formulas, I was able to duplicate the answer. You should be able to also. Write if you have questions.
 Probability-and-statistics/349773: There are three barrels and each barrel contains an equal number of red, blue, green and yellow balls, The balls are buried in sawdust so that people cannot see them. You win by choosing 3 of the same color balls. You choose a ball from each barrel. \$1.00 per game. 3 balls of the same color wins \$20. 1. How many different ways are there of choosing the three balls from the barrels? Order matters. 2. How many of them are wins? 3. What is the probability of winning on one turn? 4 Do you think the game will make money? And how much will she expect to win or lose.1 solutions Answer 250015 by Theo(3458)   on 2010-09-29 17:25:10 (Show Source): You can put this solution on YOUR website!3 barrels each contains equal number of r, b, g, y balls. probability of getting all 3 of the same color is? p of getting first ball is equal to 1 p of getting second ball the same color is .25 p of getting third ball the same color is .25 probability of getting 3 of the same color is 1 * .25 * .25 = .0625 p of all red = .25 * .25 * .25 = .015625 p of all blue = .25 * .25 * .25 = .015625 p of all green = .25 * .25 * .25 = .015625 p of all yellow = .25 * .25 * .25 = .015625 p of either red or blue or green or yellow = 4 * .015625 = .0625 the total different ways of choosing a ball from each barrel would be: 4 * 4 * 4 = 64 the possible ways to win would be 4 * 1 * 1 = 4 winning means either all red, or all green, or all blue, or all yellow. the probability of getting a win would be 4/64 = .0625 the probability of winning on any one turn is .0625. if the probability of winning on any one turn is .0625, then the probability of losing on any one turn is 1 - .0625 = .9375. if the bet is 1 dollars, then every 16 times, you will win once and lose 15 times. your total loss for the 16 times at bat is 15 dollars. your total winning has to be 15 dollars for you to break even. the pot, therefore, needs to be 15 dollars in order for you to break even. mathematically that comes out to be .9375 / .0625 = 15. in English, that means the pot should be equal to the probability of losing divided by the probability of winning in order to break even. if the pot is more than that, then you have the advantage. if the pot is less than that, then the person running the game has the advantage. since, in this case, the pot is 20 dollars, you have the advantage and the person running the game has the disadvantage.
 expressions/349729: Answer the following questions in the space provided. Lisa invested in corn. She lost 5% of her investment, costing her \$110. What was her initial investment? (A full 5-STEP SOLUTION not required, but do show work. Help please?! I don't know what to do!!!1 solutions Answer 249987 by Theo(3458)   on 2010-09-29 16:26:10 (Show Source): You can put this solution on YOUR website!let I = investment. let L = .05 * I = 110 you get .05*I = 110 divide both sides of that equation by .05 to get: I = 110/.05
 Probability-and-statistics/349764: Approximately 15% of the calls to an airline reservation phone line result in a reservation being made. Suppose that an operator handles 15 calls. What is the probability that none of the 15 calls result in a reservation? (Give the answer to 3 decimals places.) 1 solutions Answer 249985 by Theo(3458)   on 2010-09-29 16:20:27 (Show Source): You can put this solution on YOUR website!probability of a call resulting in a reservation = .15 probability of a call not resulting in a reservation = .85 15 calls are made. what is the probability that at least 1 of them will be a reservation? that equals 1 minus the probability that all of them do not result in a reservation. since the probability of any one call not resulting in a reservation equals .85, then the probability that all 15 calls will not result in a reservation equals .85^15. that number is .087354219. the probability that at least 1 call will result in a reservation is 1 minus the probability that no calls will result in a reservation. that probability = .912645781. that's the probability that at least 1 call will result in a reservation.
 Systems-of-equations/349748: a science teacher has a supply of 5% and a supply of 65% hydrochloric acid (HCI) how much of each solution should the teacher mix together to get 42mL of 45%HCI for an experiment?please help me im at a loss1 solutions Answer 249984 by Theo(3458)   on 2010-09-29 16:09:57 (Show Source): You can put this solution on YOUR website!teacher has 2 solutions. 1 = 5% hydrochloric acid. 1 = 65% hydrochloric acid. how much of each solution to get 42 mL of 45% hydrochloric acid? let x = number of mL of first solution. let y = number of mL of second solution. amount of HCI in first solution is .05*x amount of HCI in second solution is .65*y teacher wants 42 mL of 45% HCI. amount of HCI in the solution that the teacher wants is .45 * 42 mL = 18.9 mL. first equation is .05*x + .65*y = 18.9 mL of HCI. first equation tells teacher how much HCI in each solution is required to total the amount of HCI required in the final solution. second equation is x + y = 42 mL of total solution. second equation tells teach how much total of each solution is required to get the proper mix. these 2 equations need to be solved simultaneously to get the right answer. the 2 equations are: .05*x + .65*y = 18.9 x + y = 42 solve for y in the second equation to get y = 42 - x. substitute for y in the first equation to get: .05*x + .65*(42-x) = 18.9 first equation now has 1 unknown and can be solved. solve for x. simplify by removing parentheses in second equation to get: .05*x + .65*42 - .65*x = 18.9 simplify further to get: .05*x + 27.3 - .65*X = 18.9 subtract 27.3 from both sides of the equation to get: .05*x - .65*x = - 8.4 combine like terms to get: -.6*x = -8.4 divide both sides of the equation to get: x = -8.4/-.6 = 14. since x + y = 42, then 14 + y = 42. subtract 14 from both sides of this equation to get: y = 42 - 14 = 28 teacher has x = 14 and y = 28 which means that: teacher needs 14 mL of the 5% solution and 28 mL of the 65% solution. if teacher does that, then teacher gets: .05*14 + .65*28 = .7 + 18.2 = 18.9 mL of HCI. Answer looks good. Answer is: teacher needs 14 mL of 5% solution and 28 mL of 65% solution to get 42mL of 45% solution. 14 + 28 = 42 mL total solution. .05*14 + .65*28 = 18.9 mL of HCI. 18.9 / 42 = .45 = 45% of HCI in final solution.
 Graphs/349529: Beth has two part-time jobs: tutoring and law mowing. She wants to work NO MORE than 12 hours per week; at least 4 hours tutoring and not more than 6 hours mowing. She makes \$5/hour tutoring and \$6.50/hour mowing. What combination of hours would make her the most money? Identify the constraints: Graph this on an x-y axis Write an objective function1 solutions Answer 249856 by Theo(3458)   on 2010-09-29 08:04:20 (Show Source): You can put this solution on YOUR website!let x = hours spent tutoring. let y = hours spend lawn mowing. x + y < 12 is one equation. x >= 4 is another equation. y <= 6 is another equation. y >= 0 is another equation (number of hours can't be negative). All 4 are constraint equations. The equation she is looking to maximize is the revenue equation. That equation is R = 5*x + 6.5*yt She needs to graph the constraint equations and then look at the intersection points to find the maximum revenue. To graph the equations, make them an equality rather than an inequality. you get: ```x + y = 12 (E2 in the picture) x = 4 (E1 in the picture) y = 6 (E3 in the picture) y = 0 (E4 in the picture) ``` x + y = 12 is equivalent to y = 12 - x when you solve for y. replace x + y = 12 with y = 12 - x and your equations are: ```y = 12 - x (E2 in the picture) x = 4 (E1 in the picture) y = 6 (E3 in the picture) y = 0 (E4 in the picture) ``` your intersection points for these 4 constraint equations are: (x,y) = : (4,0) (4,6) (6,6) (12,0) Solve the revenue equation at these intersection points. revenue equation is: R = 5*x + 6.5*y at (4,0) this equation becomes 5*4 + 6.5*0 = 20 at (4,6) this equation becomes 5*4 + 6.5*6 = 59 at (6,6) this equation becomes 5*6 + 6.5*6 = 69 at (12,0) this equation becomes 5*12 + 6.5*0 = 60 maximum revenue occurs at (6,6). 6 hours of tutoring and 6 hours of mowing. it meets the constraints since hours of tutoring is greater than or equal to 4 and hours of mowing is less than or equal to 6 and total hours is less than or equal to 12. obviously, if hours of mowing was allowed to be more than 6, she would spend more hours mowing because mowing gives her 6.5 per hours while tutoring only gives her 5, but the constraints forced hours of mowing to be equal to or less than 6. note that the maximum hours allowed for mowing is the maximum revenue solution. this won't always be the case, but it is here because there are no other constraints that would not make it so. the picture of the constraint equations is shown below. the shaded area is the area in which all constraint are met.
 Geometry_Word_Problems/349565: A large wheel and smaller wheel interlocking .The smaller wheel radius 5.23 cm and large wheel radius 8.16 cm.The rotation of the smaller wheel causes the larger wheel to rotate. Through how many degrees will the larger wheel rotate if the smaller one rotates through 60.0 degrees ? 1 solutions Answer 249841 by Theo(3458)   on 2010-09-29 06:28:44 (Show Source): You can put this solution on YOUR website!smaller wheel radius = 5.23 cm. larger wheel radius = 8.16 cm. circumference of smaller wheel = 2*pi*5.23 = 32.86105916 cm. circumference of larger wheel = 2*pi*81.16 = 51.27079211 cm. for each revolution of the smaller wheel, the larger wheel will go through .640931373 of a revolution. when the smaller wheel rotates 60 degrees, it has rotated 1/6 of its circumference. 32.86105916 / 6 = 5.4767843193 cm. the circumference of the larger wheel will have moved the same distance. 5.4767843193 / 51.27079211 = .106821895 which means that the larger wheel has moved through 1.06821895 of its circumference. that would be equal to approximately 360 * .106821895 = 38.45588235 degrees. since the number of degrees the wheel turns is equivalent to the proportion of the circumference that it moves, then you can shorten this equation to become the ratio of the circumference of the smaller wheel to the circumference of the larger wheel. that ratio is equal to 32.86105916 / 51.27079211 = .640931373. if the smaller wheel moves through a rotation of 60 degrees, then the larger wheel move through a rotation of .640931373 * 60 = 38.45588235 degrees. 60 degree rotation in the smaller wheel = 60 / 360 * 32.86105916 = 5.476843193. 38.45588235 degree rotation in the larger wheel = 38.45588235 / 360 * 51.27079211 = 5.476843193. concepts: circumference of circle = 2 * pi * r length of arc of circle = degrees of rotation / 360 * 2 * pi * r full rotation = 360 / 360 * 2 * pi * r = circumference of the circle. wheels interlocking means a movement of the circumference of the smaller wheel is exactly equal to the movement of the circumference of the larger wheel. the movement is the same in both wheels, but the proportion of movement in each wheel is different. to see clearer, let the circumference in the smaller wheel be 6 and the circumference in the larger wheel = 12. they are interlocking. the smaller wheel rotates through 360 degrees. circumference of smaller wheel moves 6 which is one complete rotation of the smaller wheel. circumference of larger wheel also moves 6 only this is only 1/2 the complete rotation of the larger wheel.
 Travel_Word_Problems/349523: how many minutes will it take to travel 2.5 miles if you are walking at 3.7 miles per hour?1 solutions Answer 249839 by Theo(3458)   on 2010-09-29 06:05:27 (Show Source): You can put this solution on YOUR website!rate * time = distance. divide both sides of this equation by rate to get: time = distance / rate. distance = 2.5 miles rate = 3.7 miles per hour. time = 2.5 / 3.7 = .675675676 hours 60 minutes per hour makes .675675676 hours = 40.54054054 minutes.
 Linear-equations/349454: Is there a difference between solving a system of equations by the algebraic method and the graphical method? Why or why not?1 solutions Answer 249834 by Theo(3458)   on 2010-09-29 04:14:41 (Show Source): You can put this solution on YOUR website!the graph will give you an approximate solution. the algebra will give you an exact solution. the reason for this is that the accuracy of the graph is not sufficient to home in on the exact solution. an example: x^2 + y^2 = 64 y = 4 solve this algebraically and you will see that the solution is: x = +/- 6.92820323 solve this graphically and you will see that the solution lies somewhere around that value, but there's no way you can get it right on unless you zero in to a very minute detail which is not possible with most graphing software that might be available to you. the exact solution is subject to the vagaries of the accuracy of the graphing software that you use. here's the solution on the algebra.com graphing software. you can see that the points of intersection are somewhere around x = +/- 7 but there's no way you can see that the actual solution is +/- 6.92820323. The graphing solution is good to give you a general idea of where your solution lies, or if you even have a solution. To get the exact solution, you need to go to the algebraic solution.
 Linear-equations/349550: How do i graph y<-2x-1?1 solutions Answer 249832 by Theo(3458)   on 2010-09-29 04:01:34 (Show Source): You can put this solution on YOUR website!you graph the equation of y = -2x - 1. you shade the area of y < -2x - 1. it will be the area of the graph that is underneath the line of the equation. the graph of the equation of y = -2x - 1 will look like this. you can see that, for every value of x, y = -2x-1 will be on the line and y < -2x-1 will be underneath the line.
 Functions/349556: I have ran into a synthesis problem that I cannot seem to answer as I feel that the problem has not given me enough information to complete what was given to me as a problem to complete. Let and . Find and .1 solutions Answer 249831 by Theo(3458)   on 2010-09-29 03:57:25 (Show Source): You can put this solution on YOUR website!your equations are: f(x) = 3x^2 - 1 g(x) = 2x + 5 you want to find f(g(-1)) first you have to find g(-1) since g(x) = 2x + 5, then g(-1) = 2*(-1) + 5 = -2 + 5 = 3 f(g(-1)) becomes f(3) because g(-1) = 3 since f(x) = 3x^2 - 1, then f(3) = 3*(3^2) - 1 = 3*9 - 1 = 27 - 1 = 26 solve for the inner function first and then use that value to solve for the outer function. you want to solve for g(f(-1)) since f(x) = 3x^2 - 1, then f(-1) = 3*(-1)^2 - 1 = 3 - 1 = 2 since g(x) = 2x + 5, then g(f(-1)) = g(2) = 2*2 + 4 = 4 + 5 = 9
 test/349544: How would you make this problem into an algebraic expression? 48 less a number e1 solutions Answer 249830 by Theo(3458)   on 2010-09-29 03:37:31 (Show Source): You can put this solution on YOUR website!48 less than e would be written as: e - 48 48 less a number e would be written as: 48 - e
 Equations/349546: How would you solve 51=m-40?1 solutions Answer 249829 by Theo(3458)   on 2010-09-29 03:35:08 (Show Source): You can put this solution on YOUR website!equation is: 51 = m - 40 add 40 to both sides of the equation to get: 91 = m
 Probability-and-statistics/349555: 70% of people liked a hamburger. on a certain day 8 people bought it for the first time. What is the expected number of people who will like the hamburger ?1 solutions Answer 249828 by Theo(3458)   on 2010-09-29 03:33:38 (Show Source): You can put this solution on YOUR website!70% of 8 = .7 * 8 = 5.6 5.6 people will like the hamburger. this means that the expectation is that 5 or 6 people will like the hamburger, with a slight edge given to 6 people liking it rather than 5.
 Linear-systems/349538: Part 1: Suppose you need to solve a system of equations in which one equation represents a circle and the other represents a line. How many solutions can your system have? Select ALL that are possible. 0,1,2,3,4, infinitely many Part 2: How many solutions does the following system of equations have? x2+y2=64 x=1 0,1,2,3,4, infinitely many1 solutions Answer 249827 by Theo(3458)   on 2010-09-29 03:26:23 (Show Source): You can put this solution on YOUR website!one equation represents a circle and the other equation represents a line. you can have 0, 1, or 2 solutions. 0 if the line and the circle never intersect 1 if the line is tangent to the circle. 2 if the line intersects the circle at any other angle. the line can never intersect the circle at more than 2 points. your equation is: x^2 + y^2 = 64 x = 1 this is like solving equations simultaneously to get a common solution. if x = 1, you can substitute in the first equation to get 1 + y^2 = 64 which give you y^2 = 63 which gives you y = +/- sqrt(63). this implies 2 solutions to this equation. The graph of the equation of the circle should confirm this. To graph the equation of the circle, solve for y to get: y = +/- sqrt(64-x^2) that graph is shown below: Draw a vertical line at x = 1 and you will see that the intersection of that vertical line and the circle is around y = +/- 7.937253933. I drew 2 horizontal lines at y = +/- 7.937253933 so you can see that a little easier. when x = 0, y = +/- sqrt(64-0) = +/- sqrt(64) = +/- 8. That's the highest and lowest point of the circle. when x = 1, y = +/- sqrt(64-1) = +/- sqrt(63) = +/- 7.937253933. That's close to the highest point but hot exactly there (just a little below). Solving the 2 equations simultaneously yielded the result.
 Linear-systems/349098: How do I graph 2y-5x<131 solutions Answer 249524 by Theo(3458)   on 2010-09-28 09:16:59 (Show Source): You can put this solution on YOUR website!you need to solve for y. then you can graph it. 2y-5x < 13 add 5x to both sides to get: 2y < 5x + 13 divide both sides by y to get: y < (5x+13)/2 now you can graph the equation of: y = (5x+13)/2. any value of y below the the graph of that equation is good. any value of y on or higher than the graph of that equation is not good. you would shade the area of the graph below the line to indicate that is the good region. your graph would look like this: to prove your graph is good, take the value of x = 0 and substitute it in your equation to get: y = (5x+13)/2 which becomes y = 13/2 which becomes y = 6.5 which is confirmed by the graph of the equation crossing the y-axis at x = 6.5. to further prove your graph is good, take the value of y = 0 and substitute it in your equation to get: 0 = (5x+13)/2. multiply both sides of the equation to get: 0 = 5x + 13 subtract 13 from both sides of the equation to get: -13 = 5x divide both sides of the equation by 5 to get: -13/5 = x -13/5 is equal to -2.6 which is the value of y when x = 0 and is confimred by the graph crossing the x-axis at approximately that point. you don't graph the inequality directly. you graph the equality and then apply logic so see where the values of y can lie. in this case the values of y that lie below the equation are the values that are valid. take any value of x. find the value of y that lies on the graph of the equation of y = (5x+13)/2. for that value of x, any value of y less than that will be below the line, and any value of y greater than that will be above the line. the region that is valid for the equation y < (5x+13)/2 will be below the line.
 Rectangles/349086: I Believe Its 17 But Im Not Shure. Please Help? A rectangle has a length of 15 feet and a width of 8 feet. How long is a diagonal of this rectangle?1 solutions Answer 249515 by Theo(3458)   on 2010-09-28 08:59:43 (Show Source): You can put this solution on YOUR website!the diagonal forms the hypotenuse of a right triangle. the length is one leg of the right triangle. the width is the other leg. if you let: a = 1 leg of the right triangle b = the other leg of the right triangle c = the hypotenuse of the right triangle then: c^2 = a^2 + b^2 which becomes: c^2 = 15^2 + 8^2 which becomes: c^2 = 225 + 64 which becomes: c^2 = 289. Take the square root of 289 to get: c = 17. You got it right.
 Inequalities/349087: I am trying to solve the below problem but I have question. -7(5y – 2) > 6(6y-1) I first multiplied -35y- -14> 36y-6 Then added 6 to both sides -35y- -14+6>36y -6+6 -35y- -8>36y I am confused, I am not sure if the next step is correct. 1 solutions Answer 249513 by Theo(3458)   on 2010-09-28 08:56:36 (Show Source): You can put this solution on YOUR website!I will solve it for you and then you can see where you went wrong. problem is: -7(5y – 2) > 6(6y-1) which can also be written as: -7*(5*y – 2) > 6*(6*y-1). The * indicates multiplication and can make things clearer if there is any ambiguity in how the formula is shown. remove parentheses to get: -35*y + 14 > 36*y - 6 add 6 to both sides of the equation to get: -35*y + 20 > 36*y add 35*y to both sides of the equation to get: 20 > 36*h + 35*y combine like terms to get: 20 > 71*y divide both sides of the equation by 71 to get: 20/71 > y if 20/71 > y, then: y < 20/71 that should be your answer. you confirm by substituting in the original equation to see if the equation is true. take 3 values for y. pick values that are easy to calculate wherever possible. take y = 0 which is smaller than 20/71 take y = 20/71 which is equal to 20/71 take y = 1 which is greater than 20/71 if y is smaller than 20/71, the original equation should be true. if y is greater than or equal to 20/71, the original equation should be false. the original equation is: -7(5y – 2) > 6(6y-1) substitute 0 for y to get: 14 > -6 which is true. substitute 1 for y to get: -21 > 30 which is false. substitute 20/71 for y to get: 4.14084507 > 4.14084507 which is false. confirmation of y < 20/71 is good since: when y < 20/71, the original equation is true. when y >= 20/71, the original equation is false.
 Polynomials-and-rational-expressions/348066: What is the least common multiple of 3w^4x^6y and 16w^3x^6 1 solutions Answer 248899 by Theo(3458)   on 2010-09-26 03:59:11 (Show Source): You can put this solution on YOUR website!A common factor of both expressions would probably be w^3*x^6 Your first expression would be equivalent to (w^3*x^6)*(3*w*y) = 3*w^4*x^6*y Your second expression would be equivalent to (w^3*x^6)*(16) = 16*w^3*x^6 To confirm this is true, then use some random numbers for w and x and y and see if the relationships are equivalent. Let w = 3 and x = 4 and y = 5. (w^3*x^6) = 110592 (w^3*x^6) * (3*w*y) becomes 110592 * 45 = 4976640 (w^3*x^6) * (16) becomes 110592 * 16 = 1769472 3*w^4*x^6*y = 4976640 which makes it equivalent to (w^3*x^6) * (3*w*y) 16*w^3*x^6 = 1769472 which makes it equivalent to (w^3*x^6) * (16) The common multiple appears to be: w^3*x^6 I couldn't find any other common multiples in these equations. The applicable basic laws of multiplication and exponentiation appear to be: a*(b*c) = (a*b)*c a*b = b*a a^b * a^c = a^(b+c)
 Sequences-and-series/346582: Hello I seem to be stuck at this problem. I'm not to sure how to write it as the n and n-1 are subscripts. Show that an = Sn – Sn-11 solutions Answer 247769 by Theo(3458)   on 2010-09-22 08:04:59 (Show Source): You can put this solution on YOUR website!You can use brackets, or you can use parentheses. parentheses are easier to write. brackets are more formal. i'll use parentheses. formula is a(n) = S(n) - S(n-1) where () means sub. In general, S(n) = a(1) + a(2) + ........ a(n-1) + a(n) In general S(n-1) = a(1) + a(2) + ........a (n-1) If you subtract one from the other, you are left with a(n) because all of the a's from a(0) to a(n-1) cancel out. a(1) - a(1) = 0 a(2) - a(2) = 0 a(n-1) - a(n-1) = 0 a(n) - 0 = a(n) nothing left but a(n). there's a more formal proof that can be done, but it's more complicated to go through and not worth the effort unless you absolutely have to do it that way. for your purposes, this shows that it's true and should be sufficient. you can then cover with examples: let a1 = 5 and let d = 7 (d = common difference in arithmetic sequence). s(7) = 5 + 12 + 19 + 26 + 33 + 40 + 47 = 182 s(8) = 5 + 12 + 19 + 26 + 33 + 40 + 47 + 54 = 236 s(8) - s(7) = 236 - 182 = 54
 Polynomials-and-rational-expressions/346043: Graph f(x)=-x^2+4x-3, labeling the y-intercept, vertex, and axis of symmetry1 solutions Answer 247439 by Theo(3458)   on 2010-09-21 09:14:35 (Show Source): You can put this solution on YOUR website! looks like your y-intercepts is at (0,-3). looks like your x-intercepts are at (1,0) and (3,0). looks like your axis of symmetry is at x = 2. looks like your vertex is at (2,1). coordinates are shown as (x,y) where x is the value along the x-axis and y is the value along the y-axis. the point (2,1) is the intersection of a vertical line at x = 2 and a horizontal line at y = 1. you find the x-intercepts by solving for the roots of the equation. your equation is -x^2 + 4x - 3. to find the roots you set the equation equal to 0. you get -x^2 + 4x - 3 = 0 it's easier to factor if the coefficient of the x^2 term is positive so multiply both sides of the equation by -1 to get: x^2 - 4x + 3 = 0 multiplying both sides of an equality by -1 keeps the equality so you are changing the equation, but not changing the equal relations between the 2 sides of the equation. the equal relationship is what's important to solving for the roots, so changing the equation by multiplying both sides by -1 will give you the same answer. down below, at the end, i'll show you how the factors worked out if you did not multiply both sides of the eqution by -1 before factoring. your factors appear to be (x-3) * (x-1) = 0 if you multiply (x-3) * (x-1), you will get x^2 -x -3x + 3 which simplifies to x^2 - 4x + 3. this makes the value of y equal to 0 then x = 3 or x = 1. plug 3 into your original equation and you will see that it will equal to 0. plus 1 into your original equation and you will see that it will also equal to 0. the graph confirms that the values of x are 1 and 3 when the value of y is equal to 0. since the coefficient of your x^2 term in the original equation is negative, the graph of the equation will point up and open down. that means that the vertex of the equation will be a maximum point. if the coefficient of your x^2 term was positive, the graph ofr the equation would have pointed down and open up. that means that the vertex of the equation would have been a minimum point. but it's not. it's a maximum point because the coefficient of the x^2 term is negative. the vertex of your equation is found by the equation: x = -b/2a. once your equation is in standard form, you can find the value of a, b, and c. the standard form of a quadratic equation is ax^2 + bx + c = 0 your equation, in standard form, is -x^2 + 4x - 3 = 0. that makes a = -1, b = 4, c = -3 a is the coefficient of the x^2 term. b is the coefficient of the x term. c is the coefficient of the constant term. the formula for the vertex of a quadratic equation is x = -b/2a. in your equation, that translates to x = - (-4) / 2 which becomes 4/2 which becomes 2. the y value of the vertex is the value of y when x = 2. that would be -x^2 + 4x - 3 = -(2^2) + 8 - 3 = -4 + 8 - 3 = 4 - 3 = 1. that puts the vertex of your quadratic equation at (x,1) as confirmed by the graph. the axis of symmetry of your graph is the x value of your vertex. that makes the axis of symmetry of your graph at x = 2. to prove that it is the axis of symmetry, you take a value of y and solve for x. you can choose value of y = 0 to make it easy. when y = 0, x = 1 and x = 3. that makes the value of x equidistant from the axis of symmetry as it should be. the y-intercept is the value of y when x = 0. to find that, simply replace x with 0 in your equation to get y = -x^2 + 4x - 3 becomes y = 0 + 0 - 3 which becomes y = -3 as confirmed by the graph. to plot the graph, you simply take values of x and solve for y and then plot the value of the (x,y) pairs on the graph. a simple table below shows you what you will find. ``` X Y = -X^2 + 4X - 3 -2 -15 -1 -8 0 -3 1 0 2 1 3 0 4 -3 5 -8 6 -15 ``` Since the axis of symmetry is at x = 2, you can see that: when x = 0 and x = 4, the value of y is equal to -3. that's because 0 is 2 units away from 2 and 4 is also 2 units away from 2. since the graph is symmetric about the value of x = 2, the value of y at those points equi-distant from the axis of symmetry will be the same. same goes for y = -15 when x = 6 and x = -2. same goes for y = -8 when x = 5 and x = -1. back to factoring. if you did not multiply both sides of the equation by -1, you can still factor it, only it's a little harder to see the factors. i'll do it here just to show you. your original equation is -x^2 + 4x - 3. since the coefficient of the x^2 term is negative, one of your x values has to be negative, so you will get factors that look like: (-x + a) * (x + b) = -x^2 + 4x - 3 a and b are the constant terms that you want to find. since your constant term has to be -3, then you will multiply by 3 * -1, or you will multiply by -3 * 1 let's try -3 * 1. you will get (-x - 3) * (x + 1) that results in -x^2 - x -3x - 3 which results in -x^2 -4x - 3 which is not what we want. let's try (-x + 3) * (x - 1) that results in -x^2 - x + 3x - 3 which results in -x^2 + 2x - 3 which is not what we want. let's try (-x + 1) * (x - 3) that results in -x^2 + 3x + x - 3 which results in -x^2 + 4x - 3 which is what we want so we can stop here. our factors become: (-x+1) * (x-3) = 0 from the factor of (x-3) = 0, we derive x = 3. frm the factor of (-x+1) = 0, we derive -x = -1 which results in x = 3 when we multiply both sides of the equation by -1. whether we multiplied both sides of the quadratic equation in standard form by -1 in the beginning or not, we still wind up with the same factors. it's just easier to see what the factors are by doing that. you are left to do that or not at your discretion. i always found it easier to solve the quadratic equation if the coefficient of the x^2 term was positive.
 Probability-and-statistics/344412: Three machines A, B, and C are used to produce the same part, and thier outputs are collected in a single bin. Machine A produces 26% of the parts in the bin, B, 38% and machine C the rest. Of the parts produced by machine A, 8% are defective. Similary 5% of the parts from B and 4% of the parts from C are defective. Now how do you draw a probability tree to depict this situation. B. If a part is chosen randomly from the bin, if a part is defective, what is the probability that it was produced by machine A, and if the part is good, what is the probability that it was produced by machine B.1 solutions Answer 246343 by Theo(3458)   on 2010-09-17 05:41:12 (Show Source): You can put this solution on YOUR website!Your tree diagram would look something like this: ``` A B C x x x x x x x x x x x x =.26*T =.38*T =.36*T xxxxxxxxx xxxxxxxxx xxxxxxxxx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x =.08*A =.92*A =.05*B =.95*B =.04*C =.96*C ``` T = total number of parts in the bin. A = total number of parts contributed by machine A. B = total number of parts contributed by machine B. C = total number of parts contributed by machine C. Since A = .26 * T and B = .38 * T and C = .36 * T, then: Total defective in the bin from machine A would be .26 * .08 = .0208 * T Total defective in the bin from machine B would be .38 * .05 = .0190 * T Total defective in the bin from machine C would be .36 * .04 = .0144 * T Total defective parts in the bin is equal to (.0208 + .0190 + .0144) * T. Total defective parts in the bin is therefore equal to .0542 * T. Similarly, Total good in the bin from machine A would be .26 * .92 = .2392 * T Total good in the bin from machine B would be .38 * .95 = .3610 * T Total good in the bin from machine C would be .36 * .96 = .3456 * T Total good parts in the bin is equal to (.2392 + .3610 + .3456) * T. Total good parts in the bin is therefore equal to .9458 * T. The total good and bad in the bin should be equal to T. .0542 * T + .9458 * T = 1 * T = T so this is ok. Given that you pick a defective part out of the bin, what is the probability that it came from machine A? The probability that it came from machine A would be .0208 / .0542 = .383763838. Given that you pick a good part out of the bin, what is the probability that it came from machine B? The probability that it came from machine B would be .3610 / .9458 = .38168746.