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# Recent problems solved by 'oberobic'

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 Human-and-algebraic-language/272561: The Bon Appetit Bakery makes 4 1/2 times as much revenue on donuts as muffins. If total sales were \$44,000 for May, what dollar amount of each was sold?1 solutions Answer 199340 by oberobic(2304)   on 2010-02-21 21:43:20 (Show Source): You can put this solution on YOUR website!x = donut revenue y = muffin revenue x + y = 44000 so y = 44000 - x . x = 4.5*y . substituting x = 4.5(44000-x) multiply though x = 198000 - 4.5x add 4.5x to both sides 5.5x = 198000 divide both sides by 5.5 x = 36000 . y = 44000 - 36000 = 8000 . Answer: donut revenue = \$36,000 muffin revenue = \$8,000
Quadratic_Equations/272571: Find the roots of the quadratic equation x2 − 18x + 18 = 0 .
1 solutions

Answer 199338 by oberobic(2304)   on 2010-02-21 21:34:23 (Show Source):
You can put this solution on YOUR website!
x^2 - 18x + 18 = 0
 Solved by pluggable solver: SOLVE quadratic equation with variable Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=252 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 16.9372539331938, 1.06274606680623. Here's your graph:

 Coordinate-system/272576: using substitution to solve each system of quations for x=8y-10 x=4y-51 solutions Answer 199337 by oberobic(2304)   on 2010-02-21 21:24:35 (Show Source): You can put this solution on YOUR website!x=8y-10 x=4y-5 . x - 8y = -10 x - 4y = -5 subtract -4y = -5 y = 5/4 . substituting x = 8y -10 x = 8(5/4) - 10 = 0 . Solution: (0, 5/4) . Check by inspecting the graphs:
 Mixture_Word_Problems/272568: in a chemistry class, a 4% silver iodide solution must be mixed with a 10% solution to get 12 liters of a 6% solution. how many liters of the 10% solution are needed?1 solutions Answer 199335 by oberobic(2304)   on 2010-02-21 21:03:22 (Show Source): You can put this solution on YOUR website!Solution problems are solved most easily by keeping track of the amount of 'pure' stuff in the constituent solutions and in the final solution. . Here you want 12 liters of a 6% solution of silver iodide. That means you need .06*12 = .72 liters of 'pure' silver iodide. . You have a 4% and a 10% solution to add together. 'x' can be used to indicate the amount of 4% solution. We could say 'y' is the amount of 10% solution, but it is likely easier to indicate '12-x' as the amount of 10% solution. . We can now setup the equation to be solved: .04x + .10y = .72 or .04x + .10(12-x) = .72 .04x + 1.2 - .1x = .72 -.06x = -.48 x = 8 . y = 12-8 = 4 . So, our candidates are: x = 8 liters of 4% solution y = 4 liters of 10% solution . But we have to check to know if that's right...always... . How much silver iodide is there in 8 liters of 4% solution? .04*8 = .32 liters How much silver iodide is there in 4 liters of 10% solution? .1*4 = .4 liters Total = .72 liters of silver iodide . That matches what we determined we needed initially. . Checking the question, we can see the answer is only "how many liters of the 10% solution"? 4 liters
 Human-and-algebraic-language/272475: a 25 foot wire is to be cut so the longer piece is 1 foot longer than 5 times the shorter piece. find both lengths1 solutions Answer 199314 by oberobic(2304)   on 2010-02-21 18:51:26 (Show Source): You can put this solution on YOUR website!The 25-ft wire is cut into two pieces. We can call these 'x' and 'y'. And in algebra, we'd say: x + y = 25 or y = 25-x . We also are told there is a relationship between the lengths of 'x' and 'y'. x = 5y + 1 :: 'x' is 1 foot longer than 5 times 'y' . substituting y = 25-x, we have x = 5(25-x) + 1 . multiplying through x = 125 - 5x + 1 . collecting terms 6x = 126 . dividing by 6 x = 126/6 = 21 . Since x+y = 25, then y = 25 - 21 = 4 . Checking our work . Do they sum to 25? x + y = ?? 21 + 4 = 25 Yes . Is x = 5*y + 1? 5(4) + 1 = 21 ?? Yes . So our answer is: x = 21 feet y = 4 feet . Done
 Numbers_Word_Problems/272464: There is only one positive integer that is exactly twice the sum of its digits. What is this two digit number?1 solutions Answer 199308 by oberobic(2304)   on 2010-02-21 18:41:05 (Show Source): You can put this solution on YOUR website!When we present numbers we usually do not think about the deeper meaning of what they represent. We simply 'know' that 23 is twenty-three. We think the symbol '23' is a unitary thing. But, in fact, the numbers have place values such that we can say '23' means 2*10 + 3*1. So, the '2' and '3' are simply standing next to one another, they are not multiplied. . Now let's think of the number 'xy'. As a student of algebra, you doubtless think of this as x times y. That is exactly the thinking that will trip you up with number problems like these. Continuing the above example, we could say 'xy' = 23, which means 'x' is 2*10 and y = 3*1. The letters 'x' and 'y' are simply standing beside one another. . So we are told there is a positive integer that has two digits. We can call it 'xy'. And we are told that the value of 'xy' is exactly twice the sum of digits. . Well, the sum of digits in the number 'xy' is just: x + y . The value of 'xy' is 10x + y because the 'x' represents the tens and the 'y' represents the ones. . And we are given the equation, namely the value of 'xy' is exactly twice the sum of digis. 10x+y = 2(x+y) . Multiplying through 10x + y = 2x + 2y . Subtracting y from both sides 10x = 2x + y . Subtracting 2x from both sides 8x = y . Now where do we go? . Well, we can retreat to the logic of numbers that we know. If 'x' were to be any integer greater than 1, then 'y' would be two digits. But 'y' cannot be two digits because, by definition, we are told it is a single digit. Therefore, we logically conclude x = 1 And since we have shown 8x = y then we have to conclude y = 8 . Thus 'xy' has to be 18, which means the value is 1*10 + 8*1 = 18. (Remember, they're just standing beside each other, we're not multiplying them.) . The sum of digits is 1+8 = 9. . Is the value 18 equal to twice the sum of digits? Or mathematically, does xy = 2(x+y) . 18 = 2*(1+8) = 2*9 = 18 YES! . So the only positive TWO-DIGIT integer that is exactly twice the sum of its digits is 18. . Done.
 Quadratic_Equations/272459: write a quadratic equation with integral coefficients and roots equal to 2/3 and -1/21 solutions Answer 199305 by oberobic(2304)   on 2010-02-21 18:12:21 (Show Source): You can put this solution on YOUR website!In general, a quadratic equation may be depicted as y = (x + j)(x + k), where j & k are numbers. The roots of the equation will be the value of x that make x+j = 0 and the value of x that makes x+k = 0. So, we want the roots to be 2/3 and -1/2, which means: y = (x-2/3)(x+1/2) Notice the signs are reversed so the roots will be the value indicated. Multiplying through, we have y = x^2 + 1/2x - 2/3x -2/6 y = x^2 + 3/6x - 4/6x - 2/6 y = x^2 - 1/6x - 2/6 .
 Linear_Equations_And_Systems_Word_Problems/272446: the judges at marcys skating competition gave her scores of 9.6 9.2 9.4 and 9.3 what score would the fifth judge have to give marcy to raise her average score to 9.41 solutions Answer 199300 by oberobic(2304)   on 2010-02-21 17:40:00 (Show Source): You can put this solution on YOUR website!For five judge's score to have an average of 9.4, then the total points awarded would have to equal 9.4*5 = 47. This is because the average is the sum of the scored divided by the number of scores. . The four judges have awarded: 9.6 + 9.2 + 9.4 + 9.3 points = 37.5. That means the fifth judge has to award a 9.5 points for the grand total to be 47.
 Quadratic-relations-and-conic-sections/272425: please help Parabola: y=x^2+6x+5 , give vertex & x & y axis1 solutions Answer 199298 by oberobic(2304)   on 2010-02-21 17:36:08 (Show Source): You can put this solution on YOUR website!factoring x^2 + 6x + 5, we arrive at: y = (x+1)(x+5) . This means the intercepts will be -1 and -5 . . The vertex is at x= -b/2a = -6/2 = -3. When x = -3, y = -4, which is the point: (-3,-4). . By inspecting the graph, we can see this fits.
 Linear-equations/272443: Determine whether the graphs of the lines are parallel. x + 13= y y-x = -9 What is the slope line for x + 13? What is the slope line for y - x = -91 solutions Answer 199293 by oberobic(2304)   on 2010-02-21 17:27:47 (Show Source): You can put this solution on YOUR website!You have two equations that need to put into y=mx+b format. . y = x + 13 y = x - 9 . The slope of both lines is 1, so they may be parallel or the same line. They have different y-intercepts, so they are not the same line. So they are parallel. .
 Mixture_Word_Problems/272438: How many ounces of a 20% alcohol solution must be mixed with 15 ounces of a 25% alcohol solution to make a 23% alcohol solution?1 solutions Answer 199290 by oberobic(2304)   on 2010-02-21 17:23:57 (Show Source): You can put this solution on YOUR website!In solving "solutions" problems, you have to keep track of the amount of pure stuff you need at the end. . x = ounces of 20% alcohol solution to add to the 15 oz of 25% solution . 15 oz of a 25% solution = 3.75 oz of pure alcohol . The total amount can be described 15+x ounces. . We solve the problem in terms of the amount of pure alcohol .2x + 3.75 = .23(15+x) .2x + 3.75 = 3.45 + .23x . Subtracting .2x from both sides 3.75 = .03x + 3.45 . Subtracting 3.45 from both sides .3 = .03x Multiply by 100 30 = 3x Divide by 3 10 = x x = 10 . So, you have to add 10 oz of 20% alcohol solution to 15 oz of 25% alcohol. . Always check your work. . At the end we would have 25 oz that we believe would be 23% alcohol. If that is true, then we would have: .23 * 25 = 5.75 oz of pure alcohol in the solution. . We have shown (above) that we have 3.75 oz of pure alcohol in the 15 oz of 25% solution. How many oz of pure alcohol is there in 10 oz of 20% alcohol? .2*10 = 2 oz . 3.75 + 2 = 5.75 oz, which is exactly what we needed . check the question to be sure you answer it at the end... . How many ounces do you need to add? You need to add 10 ounces of 20% alcohol solution.
 Equations/272434: gary has a total of 27 dvd's and cd's. the number of dvd's is 3 less than twice the number of cd's. how many of each does he have?1 solutions Answer 199283 by oberobic(2304)   on 2010-02-21 17:08:56 (Show Source): You can put this solution on YOUR website!x = number of dvd y = number of cd x + y = 27 x = 2y - 3 . set up a simultaneous equations x + y = 27 x - 2y = -3 subtract 3y = 30 y = 10 . so x = 27 - 10 = 17 . Answer: 10 CDs 17 DVDs . Done
 Linear_Equations_And_Systems_Word_Problems/272430: How would u solve a systems of equations applications like this one. A class divides into two groups for a science experiment. the first group has five fewer students than the second group. if there are 27 students in the class, how many students are in each group?1 solutions Answer 199281 by oberobic(2304)   on 2010-02-21 17:05:43 (Show Source): You can put this solution on YOUR website!The class has 27 students divided into two groups. x = number of students in one group y = number of students in the other group x + y = 27 describes the whole class. . One group has 5 fewer than the other. This can be shown as: x = y + 5 :: which means y has 5 fewer than x . this can be solved as a system of equations. . x + y = 27 x - y = 5 add them 2x = 32 x = 16 . which means y = 27 - 16 = 11 . Answer: One group has 16 and the other has 11. . Done.
 Polynomials-and-rational-expressions/272395: Maria can shovel snow from her driveway in 65 minutes. Tom can do the same job in 45 minutes. How long would it take Maria and Tom to shovel the driveway if they worked together?1 solutions Answer 199273 by oberobic(2304)   on 2010-02-21 16:43:01 (Show Source): You can put this solution on YOUR website!With word problems like these, you have to take into consideration that there is one job that will be done in a specific period of time. . Tom can do 1 job (the whole job) in 45 minutes. So he can do 60/45 jobs per hr. Maria can do 60/65 jobs per hour. . Working together, they can do: 60/45x + 60/65x = 1 (4/3 + 12/13)x = 1 (1.33333 + .92037)x = 1 2.25370x = 1 x = 1/2.2537 = .44371 x = .44371 hr = 26.6226 min . You can check to see if their combined work will complete in 26.6226 min. . 26.6226/45 = .591 of a job 26.6226/65 = .409 of a job . Pretty close...
 Circles/272409: a diameter of 16 centimeter has what radius1 solutions Answer 199268 by oberobic(2304)   on 2010-02-21 16:19:09 (Show Source): You can put this solution on YOUR website!The radius of a circle is one-half the diameter.
 Graphs/272186: graph -3x+4y=12 and -6x+8y=241 solutions Answer 199178 by oberobic(2304)   on 2010-02-20 23:31:13 (Show Source): You can put this solution on YOUR website!First get the equations into y = mx+ b format. . -3x +4y = 12 4y = 3x + 12 y = 3/4x + 3 . -6x +8y = 24 8y = 6x + 24 y = 6/8x + 3 = 3/4x + 3 . so you have only one line .
 Geometry_Word_Problems/272187: the difference between the length and width of a rectangle is 7 centimeters. what is the dimensions of rectangle if its perimeter is 50 centimers1 solutions Answer 199174 by oberobic(2304)   on 2010-02-20 22:58:57 (Show Source): You can put this solution on YOUR website!L = length W = width P = perimeter = 50 P = 2(L+W) = perimeter formula = 2L + 2W L - W = 7 :: given so L = W+7 . substituting 2(W+7) + 2W = 50 2W + 14 + 2W = 50 4W = 36 W = 9 . L=W+7 = 9+7 = 16 . checking 2L + 2W = 2(16) + 2(9) = 32+18 = 50 . Answer: L = 16 W = 9
 Volume/272188: A farmer has two grain silos, both shaped like right circular cylinder, with dimensions shown in the diagrams below. SILO A ^30 ft (height) SILO B ^30 ft(height) <12 ft> (width of base) <30 ft>(width of base) The farmer has the same amount of grain stored in each silo. Silo A is filled to the top. What is the height, in feet, of the level of the grain in Silo B? explain how you arrived at your answer.1 solutions Answer 199173 by oberobic(2304)   on 2010-02-20 22:54:35 (Show Source): You can put this solution on YOUR website!Silo A volume = pi*r^2*30 diameter = 12, so r = 1/2d = 6 volume = pi*36*30 = 3392.92 it is full to the top, so that is volume of grain. . Silo B has the same amount of grain in it, so it has 3392.92 cubic ft base diameter = 30 substituting the known volume 3392.92 = pi*15^2*h, where 'h' is the height of the seed in Silo B h = 3392.92/(pi*15^2) h = 4.8 ft
 Probability-and-statistics/272158: Suppose a number is chosen at random from the set {0,1,2,3,...,170}. What is the probability that the number is a perfect cube?1 solutions Answer 199172 by oberobic(2304)   on 2010-02-20 22:39:47 (Show Source): You can put this solution on YOUR website!the set {0,1,2,...,170} has 171 elements. . of these, there are only 5 perfect cubes are: 1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 5^3 = 125 . so the probability is 5/171
 Linear-equations/272095: The length of a rectangle is 5 inches more than the width. The perimeter is 34 inches. Find the length and width. Thank you.1 solutions Answer 199170 by oberobic(2304)   on 2010-02-20 22:35:11 (Show Source): You can put this solution on YOUR website!L = length W = width L = W + 5 :: given P = 2L + 2W :: known definition of the perimeter P = 34 :: given . 2L + 2W = 34 2(W+5) + 2W = 34 2W + 10 + 2W = 34 4W + 10 = 34 4W = 24 W = 6 . L = W+5 = 6+5 = 11 . checking perimeter 2L + 2W = 2(6) + 2(11) = 12 + 22 = 34 OK. . Answer: L = 11 W = 6 . Done
 Travel_Word_Problems/272177: A woman hikes to the bottom of the Grand Canyon at a rate of 4.5 mph and returns to the top by mule at a rate of 2.5 mph. If the total time of the trip is 7 hours, how lond did she walk and what was the total length of her trip? Let= statement and formula please. Thank You.1 solutions Answer 199166 by oberobic(2304)   on 2010-02-20 22:06:48 (Show Source): You can put this solution on YOUR website!d=rt, where d=distance, r=rate, and t=time t = d/r= 7 so x = time walking down @ 4.5 mph 7-x = time walking up @ 2.5 mph . total time = 7 = time walking down + time walking up . 4.5(x) = d = distance walking down 2.5(7-x) = d = distance walking up, which has to equal distance walking down . 4.5x = 2.5(7-x) = 17.5 - 2.5x adding 2.5x to both sides 7x = 17.5 x = 2.5 hr 7-x = 7 - 2.5 = 4.5 hr . so, she walks down for 2.5 hr at a pace of 4.5 mph = 11.25 miles and she walks up for 4.5 hr at a pace of 2.5 = 11.25 miles . so the total length of the trip was 11.25 + 11.25 = 22.5 miles .
 Systems-of-equations/272173: the math club and science club had fundraisers to buy supplies for a hospice. the math club spent 135 buying 6 cases of water. the science club spent 110 buying 4 cases of juice and 2 cases of water . i need 2 equations using x and y1 solutions Answer 199165 by oberobic(2304)   on 2010-02-20 21:54:31 (Show Source): You can put this solution on YOUR website!x = price of a case of water y = price of a case of juice . 6x = 135 :: 6 cases of water cost \$135 . 4y + 2x = 110 :: 4 cases of juice + 2 cases of water cost \$110 . So, there you have 2 equations using x and y. . Of course, solving this is now straightforward. x = 135/6 = 22.50 4y + 2(22.50) = 110 4y = 110 -45 = 65 y = 65/4 = 16.25 . Done
 Graphs/272170: I am trying to graph y=-5 and 5x+4y=-20 1 solutions Answer 199162 by oberobic(2304)   on 2010-02-20 21:45:24 (Show Source): You can put this solution on YOUR website!By inspection, the first equation, y = -5 is a straight line parallel to the x axis. That is, for every value of 'x', the value of 'y' is -5. . 5x + 4y = -20 rearranging 4y = -5x -20 y = -5/4x - 5 which will be a straight line sloping down from the upper left to the lower right, y intercept will be (0, -5); and when x=-4, it will cross the x axis at (-4, 0) .
 Linear-equations/272133: 2y=4x+8 x-3y= -71 solutions Answer 199161 by oberobic(2304)   on 2010-02-20 21:36:49 (Show Source): You can put this solution on YOUR website!2y = 4x +8 x - 3y = -7 . rearranging -4x + 2y = 8 x - 3y = -7 . multiply the second equation by 4 4x - 12y = -28 . so, we have -4x + 2y = 8 4x - 12y = -28 . add them -10y = -20 y = 2 . substitute 2(2) = 4x + 8 4 = 4x + 8 -4 = 4x x = -1 . checking 2(2) = 4(-1) + 8 4 = -4 + 8 = 4 TRUE . -1 -3(2) = -7 TRUE . Answer: y = 2 x = -1
 Angles/272109: find two supplementary angles such that the first angle is 9 times the second.1 solutions Answer 199151 by oberobic(2304)   on 2010-02-20 20:59:12 (Show Source): You can put this solution on YOUR website!x = 1st angle y = 2nd angle x = 9y :: given x + y = 180 :: given . substituting for x = 9y 9y + y = 180 10y = 180 y = 18 x = 9y = 9(18) = 162 . checking... x + y = 162 + 18 = 180 . Answer: x = 162 y = 18 .
 Quadratic_Equations/272157: subtract and simplify 5xsqrt18-6xsqrt21 solutions Answer 199149 by oberobic(2304)   on 2010-02-20 20:53:26 (Show Source): You can put this solution on YOUR website! Simplify the
 real-numbers/272153: find the real number solutions for the equation 3x^4+15^2-72=01 solutions Answer 199148 by oberobic(2304)   on 2010-02-20 20:41:34 (Show Source): You can put this solution on YOUR website!I assume the problem is to solve: . Factoring, we arrive at: . Real solutions will be found where or . or . There are no real solutions for this part.
 Triangles/271066: one side of a triangle is half the longest side. the 3rd side is 10 methers less than the longest side. the perimater is 53. find all sides. thank you1 solutions Answer 198478 by oberobic(2304)   on 2010-02-17 21:47:33 (Show Source): You can put this solution on YOUR website!x = longest side of triangle y = 1/2x :: y is half the longest side z = x - 10 :: z is 10 less than the longest side x + y + z = 53 . substituting x + 1/2x + x-10 = 53 . multiply everything by 2 2x + x +2x -20 = 106 . collect and simplify 5x = 126 x = 126/5 = 25.2 . y = 1/2x = 25.2/2 = 12.6 . z = x-10 = 15.2 . checking, does x+y+z = 53? 25.2 + 12.6 + 15.2 = 53 Yes! . Answer: x = 25.2 y = 12.6 z = 15.2
 Linear-systems/271058: I need help with solving linear systems. I need to use the linear combinations to solve the system of linear equations. I have tried several and they are not coming out right. 3a+9b=8b-a 5a-10b=4a-9b+5 Please help me!1 solutions Answer 198473 by oberobic(2304)   on 2010-02-17 21:36:46 (Show Source): You can put this solution on YOUR website!3a + 9b = 8b - a 5a - 10b = 4a - 9b + 5 . for each equation, collect like terms and simplify to get into the right format . 3a + 9b = 8b - a add 'a' to both sides 4a + 9b = 8b subtract 8b from both sides 4a + b = 0 . 5a - 10b = 4a - 9b + 5 subtract 4a from both sides a - 10b = -9b + 5 add 9b to both sides a - b = 5 . now both equations are in the right form 4a + b = 0 a - b = 5 . add them 5a = 5 a = 1 . substitute back . 4a + b = 0 4(1) + b = 0 4 + b = 0 b = -4 . continue checking... 3a + 9b = 8b - a ?? 3(1) + 9(-4) = 8(-4) - 1 ?? 3 -36 = -32 -1 ?? -33 = -33 YES! . 5a - 10b = 4a - 9b + 5 ?? 5(1) - 10(-4) = 4(1) - 9(-4) + 5 ?? 5 + 40 = 4 + 36 + 5 ?? 45 = 45 YES! . Answer: a = 1 b = -4
 Age_Word_Problems/271049: Trisha is 3 times as old as Lilly, as Lilly is twice as old as Emma. If the sum of the tree girls' ages is 27, how old is Emma?1 solutions Answer 198464 by oberobic(2304)   on 2010-02-17 21:13:55 (Show Source): You can put this solution on YOUR website!T = Tricia's age L = Lilly's age E = Emma's age T = 3L :: Tricia is 3 times as old as Lilly L = 2E :: Lilly is 2 times as old as Emma T + L + E = 27 :: given . substituting 3L + 2E + E = 27 3L = 3(2E) 6E + 2E + E = 27 9E = 27 E = 3 Emma is 3 years old. . L = 2E = 2(3)= 6 Lilly is 6 years old. . T = 3L = 3(6) = 18 Tricia is 18 years old. . checking...is the total 27? 3+9+18 = 27 . So the answer is correct.
 Problems-with-consecutive-odd-even-integers/271017: The sum of the squares of two consecutive positive odd integers is 130. Find the two integers1 solutions Answer 198419 by oberobic(2304)   on 2010-02-17 20:21:05 (Show Source): You can put this solution on YOUR website!x = first positive odd integer y = second positive odd integer y = x+2 = the second positive odd integer is consecutive to x. . x +y = 130 substituting x + x+2 = 130 2x + 2 = 130 subtracting 2 from both sides 2x = 128 dividing both sides by 2 x = 64, and it is NOT ODD ! So, there is no solution with odd integers. . To check, we can explore the relationship of "nearby" odd integers: x = 65 y = x+2 = 67 x+y = 132 . x = 63 y = x+2 = 65 x+y = 128 . So there is no solution with consecutive odd numbers. . If the original problem were to have been x+y = 132, then the solution would have been 65 and 67. But that was not what was given.