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# Recent problems solved by 'Theo'

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 Complex_Numbers/733391: Please solve this complex fraction. Show and explain all steps of how you solved it please. The ---- is a division line. Thank you. n - 1 ----------- 1/2 + 1/n ----------- n - 4 + 3/n ----------- 1 + 2/n Thank you for spending your time to help me.1 solutions Answer 448518 by Theo(3458)   on 2013-04-04 03:48:08 (Show Source): You can put this solution on YOUR website!see the attached worksheet and the comments below it. first you change the division into a multiplication because: (a/b) / (c/d) = (a/b) * (d/c) then you multiply the first fraction by 2n/2n and the second fraction by n/n. you are using the associative law of multiplication to do this. after this is done yuo are left with fractions that can be factored further to simplify them. once simplified, the common terms in the numerator and the denominator can be canceled out and you are left with the simplified answer. email: theoptsadc@yahoo.com website: http://theo.x10hosting.com
 Complex_Numbers/733389: Please solve this complex fraction. Show and explain all steps of how you solved it please. The ---- is a division line. Thank you. 1 + b/a ------- 1 - b/a ------- 2a/b ------- a/b - 1 Thank you for spending your time to help me.1 solutions Answer 448517 by Theo(3458)   on 2013-04-04 03:42:17 (Show Source): You can put this solution on YOUR website!see the attached worksheet and the words underneath. first you change this into a multiplication problem because: (a/b) / (c/d) = (a/b) * (d/c) then you multiply the first fraction by a/a and you multiply the second fraction by b/b. this removes the fraction parts of the numerators and denominators or each. you then remove the parts of the numerator that cancel out with equal parts in the denominator. the result is your answer. you are using the associative law of multiplication when you multiply the first fraction by a/a and the second fraction b/b. the associative law of multiplication says you can do the following: a * b * c = (a * b) * c = a * (b * c) a * b * c * d = (a * b) * (c * d)
 Complex_Numbers/733384: Please solve. Show and explain all steps of how you solved it please. Thank you. 2 + 1/x - 1/x^2 _______________ 1 + 4/x + 3/x^2 Thank you for spending your time to help me.1 solutions Answer 448515 by Theo(3458)   on 2013-04-04 03:35:54 (Show Source): You can put this solution on YOUR website!see the attached worksheet if you multiply by x^2 / x^2, you will remove the denominators and you will be left with a quadratic equation divided by another quadratic equation which, when factored, will allow some of the factors to cancel out and you are then left with the simplified solution which is your answer.
 Rational-functions/732010: Please solve and show all work. It would be greatly appreciated if you explained it too. Thanks. Divide: 4x^2 - 15 / 2x - 51 solutions Answer 447805 by Theo(3458)   on 2013-03-31 19:23:26 (Show Source): You can put this solution on YOUR website!you want to divide 4x^2 - 15 by (2x - 5) you need to fill in the missing orders of exponents. when you do that, your equation you are dividing into will be equal to: 4x^2 + 0x - 15 you want to divide that by (2x - 5) you start by dividing 2x into 4x^2 to get 2x because 2x * 2x = 4x^2 you then multiply (2x - 5) by 2x to get 4x^2 - 10x you then subtract 4x^2 - 10x from 4x^2 + 0x - 15 to get 10x - 15 you then divide 2x into 10x to get 5 because 5 * 2x = 10x you then multiply (2x - 5) by 5 to get 10x - 25 you then subtract 10x - 25 from 10x - 15 to get + 10 since + 10 is of a lower order than 2x, you are done and your remainder is 10. your quotient is 2x + 5 + 10/(2x - 5) instructions on how to do polynomial division can be found at the following website. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut36_longdiv.htm instructions on how to do synthetic division can be found at the following website. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut37_syndiv.htm email: theoptsadc@yahoo.com website: http://theo.x10hosting.com
 Rational-functions/732009: Please solve and show all work. It would be greatly appreciated if you explained it too. Thanks. Divide: 4x^2 - 3x + 2 / x1 solutions Answer 447793 by Theo(3458)   on 2013-03-31 19:02:43 (Show Source): You can put this solution on YOUR website!you are dividing 4x^2 - 3x + 2 by x divide 4x^2 by x to get 4x because x * 4x = 4x^2 subtract 4x^2 from 4x^2 - 3x + 2 to get -3x + 2 divide -3x by x to get -3 because -3 * x = -3x subtract -3x from -3x + 2 to get + 2 since x is a higher order than 2 you are done dividing and 2 is your remainder. your quotient is 4x - 3 + (2/x) if you multiply that by x you will get back to 4x^2 - 3x + 2.
 Rational-functions/725383: Please simplify this problem and give the restrictions. 8x^2 - 6x / 4x^2 Thank You for your help! :)1 solutions Answer 444597 by Theo(3458)   on 2013-03-15 08:26:24 (Show Source): You can put this solution on YOUR website!this factors to: 2x * (4x - 3) divided by 2x * 2x the 2x in the numerator and denominator cancel out and you are left with: 4x - 3 divided by 2x that's the simplified expression. the restrictions come from the original factored equation which is: 2x * (4x - 3) divided by 2x * 2x x cannot be equal to 0, otherwise the denominator becomes equal to 0. otherwise you're ok.
 Rational-functions/725385: Please Simplify and give Restrictions. x^3 + 6x^2 + 9x / x^3 - 9x Thank you for your help! :)1 solutions Answer 444593 by Theo(3458)   on 2013-03-15 08:20:32 (Show Source): You can put this solution on YOUR website!this factors to: x * (x^2 + 6x + 9) divided by x * (x^2 - 9) which factors to: x * (x + 3) * (x + 3) divided by x * (x + 3) * (x - 3) The x in the numerator and denominator cancel out. The (x + 3) in the numerator and denominator cancel out. you are left with: (x + 3) / (x - 3) that's the simplified equation. the restrictions come from the original factored equation, which is: x * (x + 3) * (x + 3) divided by x * (x + 3) * (x - 3) x in the denominator can't be equal to 0. x can't be equal to -3, otherwise (x + 3) in the denominator would be equal to 0. x can't be equal to 3, otherwise (x - 3) in the denominator would be equal to 0.
 Rational-functions/725388: Please Simplify and give Restrictions. ( x^2 + x - 6 ) ( x + 3 )^-2 Thank you for your help! :)1 solutions Answer 444591 by Theo(3458)   on 2013-03-15 08:12:46 (Show Source): You can put this solution on YOUR website!since (x+3)^-1 is equivalent to 1 / (x + 3)^2, this equation translates to: (x^2 + x - 6) divided by (x + 3)^2 this factors to: (x + 3) * (x - 2) divided by (x + 3) * (x + 3) (x + 3) in the numerator and denominator cancel out and you are left with: (x - 2) / (x + 3) that's the simplified equation. the restrictions come from the original equation after it has been factored. that equation is: (x + 3) * (x - 2) divided by (x + 3) * (x + 3) x cannot be equal to -3 because (x + 3) = 0. otherwise you're ok.
 Rational-functions/725391: Please Simplify and give Restrictions. x^2 - y^2 / x^2 + 3xy + 2y^2 Thank you for your help! :)1 solutions Answer 444590 by Theo(3458)   on 2013-03-15 08:07:43 (Show Source): You can put this solution on YOUR website!this factors to: (x-y)(x+y) divided by (x+2y)(x+y) the (x+y) in the numerator and denominator cancel out and you are left with: (x-y) divided by (x + 2y) that's the simplified expression. the restrictions come from the original equation. the original equation is: (x^2 - y^2) divided by (x^2 + 3xy + 2y^2) which factored out to be: (x - y) * (x + y) divided by (x + y) * (x + 2y) since the denominator can't be equal to 0, then the following restrictions apply. x cannot be equal to -y, otherwise (x + y) will be equal to 0. x cannot be equal to -2y, otherwise (x + 2y) will be equal to 0.
 Mixture_Word_Problems/726103: If I have 100ml of a 6% solution; how do I dilute that to a 3% solution? Thank you, chere my thought is that I would take 94% of 100ml and then add that to the 6% solution?1 solutions Answer 444471 by Theo(3458)   on 2013-03-14 13:22:31 (Show Source): You can put this solution on YOUR website!it appears that if you add 100 ml of a 0% solution, then you will have 200 ml of a 3% solution. 6% solution in 100 ml is equal to 6 ml because 6 ml divided by 100 ml = .06 which is equivalent to 6%. 6 ml divided by 200 ml is equal to .03 which is equivalent to 3%. the formula that can be used is: x = the amount of 0% solution you need to add. you get .06 * 100 + 0 * x = .03 * (100 + x) you are adding 100 ml of 6% solution to x ml of 0% solution to get a total of (100 + x) ml of 3% solution. the formula becomes: .06 * 100 + 0 * x = .03 * (100 + x) simplify to get: .06 * 100 + 0 * x = .03 * 100 + .03 * x simplify further to get: 6 = 3 + .03 * x subtract 3 from both sides of the equation to get: 3 = .03 * x divide both sides of the equation by .03 to get: 3 /.03 = x simplify to get: 100 = x. you need to add 100 ml of 0% solution to 100 ml of 6% solution to get 200 ml of 3% solution.
 Finance/726085: Find the annual perecentage yield. A bank offers an APR of 2.3% compounded daily.1 solutions Answer 444467 by Theo(3458)   on 2013-03-14 12:12:36 (Show Source): You can put this solution on YOUR website!assuming the number of days in the year is 365, the calculations are done as follows: annual percentage rate = 2.5% nominal annual interest rate = apr / 100% = .025 number of compounding periods per year = 365 daily interest rate = .025 / 365 = .00006849315068 add 1 to that to get 1.00006849315068 raise that to the 365th power to get (1.00006849315068)^365 = 1.025314243 subtract 1 from that to get .025314243 that's the effective annual interest rate. multiply that by 100% to get 2.5314243% that's the annual percentage yield. if you invest a dollar today and you get 2.5% annual percentage rate, you will have 1.025314243 dollars at the end of the year. your effective interest rate is equal to .025314243 your annual percentage yield is 2.5314243%.
 Polynomials-and-rational-expressions/726077: How do you convert y= -x +-12 from slope intercept form to standard1 solutions Answer 444462 by Theo(3458)   on 2013-03-14 11:22:15 (Show Source): You can put this solution on YOUR website!slope intercept form is y = mx + b m is the slope b is the y intercept standard form is ax + by = c a is the coefficient of x b is the coefficient of y c is the constant. to convert from slope intercept form to standard form, you need to get both the x and the y on the left side of the equation. assuming your equation is y = -x - 12, you would do the following: add x to both sides of the equation to get: x + y = -12 in this equation, that's all you have to do. you are in the form of: ax + by = c a is equal to 1 b is equal to 1 c is equal to -12
 Linear_Equations_And_Systems_Word_Problems/726065: a smoker is 15 times more likely to die of lunch cancer than a non smoker. An ex smoker who stopped smoking "t" years ago is W times more likely to die of lung cancer. w= 15-t Polly never smoked and sandy quit 2 1/2 years ago. graph the equation and use the graph to estimate how much more likely it is for sandy to die of lung cancer than Polly. 1 solutions Answer 444461 by Theo(3458)   on 2013-03-14 11:18:43 (Show Source): You can put this solution on YOUR website!Let x = the probability of a non-smoker to die of lung cancer. Let 15 * x = the probability of a smoker to die of lung cancer. Let w * x = the probability of a smoker who quit t years ago to die of lung cancer. you are given that w = 15 - t. Polly never smoked. Sandy quit 2.5 years ago. Since w = 15 - t, and t = 2.5, this means that the probability that Sandy will die of lung cancer is 12.5 * the probability that Polly will die of lung cancer. To graph the equation, let x = the number of years that the smoker has quit smoking. let y = w which is equal to 15 - x. The graph will look like this: From the graph, you can see that when x = 2.5, y = 12.5. It's actually hard to see, but you can do the calculations to prove it as follows: w = 15 - t and t = 2.5 which makes w = 15 - 2.5 which is equal to 12.5. on the graph, y is equal to w and x is equal to t. if you can graph a horizontal line at y = 12.5 (shown on the graph), and a vertical line at x = 2.5 (approximated on the graph), then the intersection of the vertical and horizontal line should be intersecting the graph of y = 15 - x at about the same point, which it does.
 Probability-and-statistics/726053: A hundred squash balls are tested from the height of the bounce. A ball is fast if it rises above 32 inches. The average height of the bounce was 30inches and the standard deviation was 3/4 inches. What is the chance of getting a fast standard ball.1 solutions Answer 444460 by Theo(3458)   on 2013-03-14 10:53:58 (Show Source): You can put this solution on YOUR website!the mean is 30 inches. the standard deviation is 3/4 inch. to find p(x > 32) you need to calculate the z-score to get: z = (32 - 30) / (3/4) which is equal to 8/3. that's a z-score of 2.67 rounding to two decimal places. the formula to find the z-score is: z-score = (value - mean) / (standard deviation) look up in the z-score tables for a z-score of 2.67 and you will find that the probability of the z-score being less than 2.67 is equal to .9962. The probability of the z-score being greater than 2.67 is equal to 1 - .9962 which is equal to .0038. the z-score table that i used is shown below: http://lilt.ilstu.edu/dasacke/eco148/ztable.htm
 Surface-area/726046: I'm trying to find the surface area of a cylinder. It has a radius of 4.7m and a height of 12m. Please help.1 solutions Answer 444456 by Theo(3458)   on 2013-03-14 10:38:13 (Show Source): You can put this solution on YOUR website!The surface area of a cylinder is equal to the lateral area plus the area of the top and bottom base. The area of each base of the cylinder is equal to pi * r^2 which is equal to pi * (4.7)^2 which is equal to pi * 22.09 which is equal to 22.09 * pi. multiply this by 2 to get a total of 44.18 * pi the lateral area of the cylinder is equal to 2 * pi * r * h which is equal to 2 * pi * 4.7 * 12 which is equal to 112.8 * pi add up 44.18 * pi and 112.8 * pi and you get a total surface area of 156.98 * pi. this is the surface area in terms of pi. multiply by the value of pi to get: total surface area equals 493.1672148 square meters which can be rounded to however many decimal places you require. here's a reference you can look at to see what these formulas mean. http://www.mathopenref.com/cylinderarea.html
 Permutations/726054: If 3 melons sell for \$0.42, how much would 7 melons cost?1 solutions Answer 444452 by Theo(3458)   on 2013-03-14 10:15:54 (Show Source): You can put this solution on YOUR website!\$0.42 * 100 = 42 cents. if 3 melons cost 42 cents, then the cost per melon is equal to 42 / 3 = 14 cents. 7 melons times the cost per melon of 14 cents would equal 7 * 14 which would be equal to 98 cents. 98 cents / 100 = \$0.98.
 test/726055: If p=19 what is 745+5p1 solutions Answer 444451 by Theo(3458)   on 2013-03-14 10:09:36 (Show Source): You can put this solution on YOUR website!if p = 19, then 745 + 5p = 745 + 5*19 = 745 + 95 = 840 you just replace p with 19 in the expression and then simplify the expression to get the answer.
 Rectangles/726045: Three multiplied by the sum of 4 and a number is the same as 18 more than the number. Find the number. If n is "the number," which equation could be used to solve for the number? 1 solutions Answer 444449 by Theo(3458)   on 2013-03-14 10:02:55 (Show Source): You can put this solution on YOUR website!let x represent the number. 3 multiplied by the sum of 4 and the number is the same as 18 more than the number. this statement translates to: 3 * (x + 4) = x + 18 solve for x as follows: simplify the equation to get: 3x + 12 = x + 18 subtract 12 from both sides of the equation to get: 3x = x + 6 subtract x from both sides of the equation to get: 2x = 6 divide both sides of the equation by 2 to get: x = 3 that should be your answer. substitute the value of 3 for the number to see if this solves the original problem. the original problem is: Three multiplied by the sum of 4 and a number is the same as 18 more than the number. Find the number. Assuming the number is 3, this statement then becomes: Three multiplied by the sum of 4 and 3 is the same as 18 more than 3. Find the number. 18 more than 3 is equal to 21. 3 multiplied by the sum of 4 and 3 is 3 multiplied by 7 which is equal to 21. This confirms the solution that the number is 3.
 Rational-functions/726020: Give (a) the domain and (b) the zeros of f. f(x)=3x^3-x^2-2x/x(x+1)^2 I would like to thank you for sparing your time to help me and if it is possible, send a response ASAP .... Thank You very much.1 solutions Answer 444447 by Theo(3458)   on 2013-03-14 09:56:06 (Show Source): You can put this solution on YOUR website!The domain appears to be all values of x except x = -1 and x = 0 The range appears to be all values of y. The zeroes (roots) appear to be x = -2/3 and x = 1. you find the domain by looking for all values of x that will result in a value of y. all values of x will result in a value of y except when x = -1 and when x = 0. when x = -1 or when x = 0, the denominator of the original equation will be equal to 0 which makes the value of y undefined. therefore x = -1 and x = 0 can't be in the domain. since there are no further restrictions as to what the value of x can be, then the domain is all values of x except at x = -1 and x = 0. the zeroes of the equation are found by setting the equation equal to 0 and solving for x. it's best to simplify the equation first. start with (3x^3 - x^2 - 2x) / (x * (x+1)^2 since (3x^3 - x^2 - 2x) / x is equal to (3x^2 - x - 2), the equation becomes: (3x^2 - x - 2) / (x+1)^2 set this equal to 0 to get: (3x^2 - x - 2) / (x + 1)^2 = 0 multiply both sides of the equation by (x + 1)^2 to get: (3x^2 - x - 2) = 0 factor to get: (3x + 2) (x - 1) = 0 solve for x to get: x = -2/3 or x = 1 these are the roots of the quadratic equation which is the point at which the equation crosses the x-axis which is the zeroes of the equation. you can confirm that the solutions are correct by graphing the equation. the graph of the equation looks like this: from this graph, you can see that the zeroes of the graph are around x = -2/3 and x = 1. from this graph, you can see that there appears to be an asymptote at around x = -1 although it's not real easy to see that. from this graph, you cannot see that there is a hole at x = 0 the value of y is undefined at x = -1 and also undefined at x = 0. this is because the denominator of the equation is 0 at those values of x. at x = -1, this results in an asymptote. at x = 0, this results in a hole. if you try to find the value of y when x = 0 and when x = -1, you will not be able to. you can find a value of y for any other value of x other than 0 or -1.
 logarithm/674335: graph this function g(x)=log2of(x+1)1 solutions Answer 419180 by Theo(3458)   on 2012-10-31 15:52:03 (Show Source): You can put this solution on YOUR website! i appears there's a vertical asymptote at x = -1. the following table helps show that. ```x y = log(2,x+1) -2 #NUM! -1 #NUM! -0.9999 -16.60964047 -0.9 -3.321928095 -0.8 -2.321928095 -0.7 -1.736965594 -0.6 -1.321928095 -0.5 -1 0 0 1 1 2 1.584962501 3 2 4 2.321928095 5 2.584962501 6 2.807354922 7 3 8 3.169925001 9 3.321928095 ``` you cannot take a log of a negative number of or 0. that is why the first 2 values in the table are invalid. that is why the graph stops as just to the right of x -1 log (x+1) will be log(0) when x is equal to -1.