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# Recent problems solved by 'EdwinParker'

 Rational-functions/625496: Find the real-number solutions of the equation. = 01 solutions Answer 393582 by EdwinParker(16)   on 2012-07-02 19:44:54 (Show Source): You can put this solution on YOUR website!``` x³ + 2x² - 25x - 50 = 0 Factor the first two terms on the left side by taking out GCF x² x²(x + 2) - 25x - 50 = 0 Now factor the last two terms on the left side by taking out GCF -25 x²(x + 2) - 25(x + 2) = 0 Now factor out the GCF (x + 2) leaving x² and -25 in parentheses: (x + 2)(x² - 25) = 0 Factor the second parentheses as the difference of squares: (x + 2)(x - 5)(x + 5) = 0 Use the zero-factor principle by setting each factor = 0: x + 2 = 0; x - 5 = 0; x + 5 = 0 x =-2; x = 5; x = -5 The real-number solutions are -5, -2, and 5. Edwin```
 Rational-functions/625451: Using multiplication solve this Rational Expression. (3z^2+12z-96)/(z^2-11z+28) * (6z^2-32z-70)/(3z^2+29z+40) Thanks,1 solutions Answer 393533 by EdwinParker(16)   on 2012-07-02 16:27:19 (Show Source):
 Finance/624014: How many years will it take \$6,000 to amount to \$8,000 if it is invested at an annual rate of 9.0% compounded continuously?1 solutions Answer 392485 by EdwinParker(16)   on 2012-06-26 14:58:51 (Show Source): You can put this solution on YOUR website!How many years will it take \$6,000 to amount to \$8,000 if it is invested at an annual rate of 9.0% compounded continuously? ```A = Pert A = final amount = 8000 P = beginning amount = 6000 r = rate expressed as a decimal = .09 t = the number of years = (unknown) 8000 = (6000)e(.09)t Divide both sides by 6000 = e.09t = e.09t Since A = eB is equivalent to B = ln(A), ,09t = ln() t = ()÷.09 t = 14.81481481... Answer: about 15 years. Edwon```
 Probability-and-statistics/616887: Hello, I'm having some problems related to test exams and probability. Say you have a test exam composed of N multiple answer questions (I have the probability of guessing each question right or wrong by chance pretty much covered, that's easy). Each question can be either correct (which will add a certain amount of points to the final score), incorrect (which will substract another quantity from the final score) or blank (which will not add nor substract). My question is: If we know the probability of guessing each question, what is the probability of achieving a certain final score guessing all the test (eg. 5.75)?1 solutions Answer 388019 by EdwinParker(16)   on 2012-06-01 14:05:23 (Show Source): You can put this solution on YOUR website!```I will assume you are implying that the student will not leave any questions blank, and will therefore either gain or lose some points for each of the N questions. let: N = the number of questions. a = the number of points added for each correct answer. b = the number of points subtracted for each incorrect answer. p = the probability of answering any one question correctly. C = the number of answers a student guesses correctly. (Therefore the student guesses N-C answers incorrectly, since I am assuming he or she leaves no answers blank.] X = the student's score Then the student's highest possible score is aN and the lowest is -bN The formula for the score X is X = aC - b(N - C) Solve that for C X = aC - bN + bC X + bN = aC + bC X + bN = C(a + b) = C C = If that does not come out to a whole number, then the student cannot possibly score exactly X, so you will then have to choose whether to round C up to the next higher integer so that he or she makes slightly higher than C or to round C down to the next lower integer and makes slightly lower than X. The probability of scoring X (or as close to X as possible) is the probability of guessing C correct answers correctly, which is: where is the number of combinations of N things taken C at a time. Edwin```
 Circles/616919: find the center of the circle using these three points (-4,4),(-7,3),(-8,2)1 solutions Answer 388005 by EdwinParker(16)   on 2012-06-01 12:12:30 (Show Source): You can put this solution on YOUR website!(-4,4),(-7,3),(-8,2) ```Substitute those three points for (x,y) into the standard equation for a circle: (x-h)² + (y-k)² = r² So we have this system of three equations in three unknowns: (-4-h)² + (4-k)² = r² (-7-h)² + (3-k)² = r² (-8-h)² + (2-k)² = r² Set the left sides of the first two equations equal: (-4-h)² + (4-k)² = (-7-h)² + (3-k)² Rearrange so as to have difference of squares on each side: (-4-h)² - (-7-h)² = (3-k)² - (4-k)² Factor both sides as the difference of squares: [(-4-h) - (-7-h)][(-4-h) + (-7-h)] = [(3-k) - (4-k)][(3-k) + (4-k)] Remove the inner parentheses: [-4 - h + 7 + h][-4 - h - 7 - h] = [3 - k - 4 + k][3 - k + 4 - k] Simplify: [3][-11 - 2h] = [-1][7 - 2k] -33 -6h = -7 + 2k -6h - 2k = 26 ------------------------------------ Set the left sides of the first and third equations equal: (-4-h)² + (4-k)² = (-8-h)² + (2-k)² Rearrange so as to have difference of squares on each side: (-4-h)² - (-8-h)² = (2-k)² - (4-k)² Factor both sides as the difference of squares: [(-4-h) - (-8-h)][(-4-h) + (-8-h)] = [(2-k) - (4-k)][(2-k) + (4-k)] Remove the inner parentheses: [-4 - h + 8 + h][-4 - h - 8 - h] = [2 - k - 4 + k][2 - k + 4 - k] Simplify: [4][-12 - 2h] = [-2][6 - 2k] -48 - 8h = -12 + 4k -8h - 4k = 36 ---------------------------------- Now we have the system of equations: -6h - 2k = 26 -8h - 4k = 36 Solve that system and get (h,k) = (-4,-1) That's the center. That's all you asked for. But if you had been asked for the equation, or if you wanted to check your problem, you would also need to find the radius. Substitute (h,k) = (-4,-1) in any one of the original 3 equations. I'll pick the first one: (-4-h)² + (4-k)² = r² (-4-(-4))² + (4-(-1))² = r² (-4+4)² + (4+1)² = r² 0² + 5² = r² 5² = r² 5 = r So the radius is 5, and so the equation of the circle is: (x+4)² + (y+1)² = 5² (x+4)² + (y+1)² = 25 Edwin```
 Miscellaneous_Word_Problems/616847: In a class, 25 students have been on a plane, 20 on a train, 27 on a boat, 9 on a plane and train, 12 on a train and boat, 14 on a plane and boat, 3 on all three, and 1 on none of them How many students are in the class?1 solutions Answer 387992 by EdwinParker(16)   on 2012-06-01 10:40:18 (Show Source): You can put this solution on YOUR website!In a class, 25 students have been on a plane, 20 on a train, 27 on a boat, 9 on a plane and train, 12 on a train and boat, 14 on a plane and boat, 3 on all three, and 1 on none of them How many students are in the class? ``` Let P = the set of all students who have been on a plane, regardless of whether they have been on a train or a boat. Let T = the set of all students who have been on a train, regardless of whether they have been on a plane or a boat. Let B = the set of all students who have been on a boat, regardless of whether they have been on a plane or a train. There are 8 regions which I have labeled with arbitrary small letters. [I skipped little "b" because I used capital "B" for the set of all students who have been on a boat.] The most inclusive clue we have is this one: ``` >>...3 on all three,...<< ``` Those 3 are in all three circles, and the only region that is common to all three circles is the middle region f, so we put 3 for f The most EXcluding clue we have is this one: ``` >>...and 1 on none of them...<< ``` That 1 is outside all three circles, and the only region outside all three circles is region i, so we put 1 in region i: Now we look at this: ``` >>...9 on a plane and train,...<< ``` 3 of those 9 are already accounted for in the middle region, so that leaves 9-3 or 6 to go in the other region common to circles P and T, which is region c, so we put 6 in that region: Now we look at this: ``` >>...12 on a train and boat,...<< ``` 3 of those 12 are already accounted for in the middle region, so that leaves 12-3 or 9 to go in the other region common to circles T and B, which is region g, so we put 9 in that region: Now we look at this: ``` >>...14 on a plane and boat,...<< ``` 3 of those 14 are already accounted for in the middle region, so that leaves 14-3 or 11 to go in the other region common to circles P and B, which is region e, so we put 11 in that region: Now we look at this: ``` >>...25 students have been on a plane,...<< ``` We have numbers in 3 of the regions of circle P, so 6+3+11 or 20 of those 25 are already accounted for so that leaves 25-20 or 5 to go in the remaining region of circle P, region a, so we put 5 in that region: Now we look at this: ``` >>...20 on a train,...<< ``` We have numbers in 3 of the regions of circle T, so 6+3+9 or 18 of those 20 are already accounted for so that leaves 20-18 or 2 to go in the remaining region of circle T, region d, so we put 2 in that region: Now we look at this: ``` >>...27 on a boat,...<< ``` We have numbers in 3 of the regions of circle B, so 11+3+9 or 23 of those 27 are already accounted for so that leaves 27-23 or 4 to go in the remaining region of circle B, region h, so we put 4 in that region: We are now ready to look at the question, for now we can answer anything we are asked because we have the complete Venn diagram: ``` >>...How many students are in the class?...<< ``` Just add up all the 8 numbers in the regions: 5+6+2+11+3+9+4+1 = 41 Answer: There are 41 students in the class. Edwin```
 real-numbers/616908: is 2.86 a rational number1 solutions Answer 387980 by EdwinParker(16)   on 2012-06-01 09:41:34 (Show Source): You can put this solution on YOUR website!```A RATIOnal number is the RATIO of two integers. 2.86 means which means which is the RATIO of two integers 286 and 100. [And yes, although irrelevant, it reduces to which is also the RATIO of two integers.] A RATIOnal number is the RATIO of two integers, therefore 2.86 is a RATIOnal number. Edwin```
 Surface-area/616393: zoe the goat is tied to a rope to one corner of a 15 meter by 25 meter rectangular barn. over what area of the field can zoe graze is the rope is 1. 10 meters long? 2. 20 meters long? 3. 30 meters long? 4. zoe is happiest when she has at least 400 meters squared to graze. what possible lengths of rope could be used?1 solutions Answer 387673 by EdwinParker(16)   on 2012-05-30 21:08:41 (Show Source): You can put this solution on YOUR website!zoe the goat is tied to a rope to one corner of a 15 meter by 25 meter rectangular barn. over what area of the field can zoe graze is the rope is 1. 10 meters long? 2. 20 meters long? 3. 30 meters long? 4. zoe is happiest when she has at least 400 meters squared to graze. what possible lengths of rope could be used? 1. 10 meters long? ``` That's 3/4ths of a circle with a radius of 10. The area of a full circle with radius 10 is A = = = Multiply that by 3/4ths gives · = = 235.6 m² --------------------------------------------- ``` 2. 20 meters long? ``` That's 3/4ths of a circle with a radius of 20. plus 1/4th of a circle with a radius of 5 The area of a full circle with radius 20 is A = = = Multiply that by 3/4ths gives · = = 942.48 m² The area of a full circle with radius 5 is A = = = Multiply that by 1/4th gives · = = 19.63 m² Total area = 942.48+19.63 = 962.11 m². ----------------------------------------------------------- ``` 3. 30 meters long? ``` That's 3/4ths of a circle with a radius of 30. ```
 Probability-and-statistics/616170: a fair six-sided die is rolled three times. what is the probability that the sum of the second and third rolls is less than the value obtained on the first roll?1 solutions Answer 387577 by EdwinParker(16)   on 2012-05-30 14:56:27 (Show Source): You can put this solution on YOUR website!```A probability has a numerator and a denominator. The denominator is easy. 6 ways for the 1st roll times 6 ways for the 2nd roll times 6 ways the 3rd roll. That's 6·6·6 = 6³ = 216 The numerator is not so easy. The smallest sum the 2nd and 3rd rolls can have is 2, and the largest sum the 2nd and 3rd rolls can have is 5, when the 1st roll is 6. So we make a chart: A B C D E F Possible Number Possible Number of Product Sum of 2nd & 3rd of ways 1st rolls possible of 2nd & 3rd rolls to to get for that 1st rolls columns rolls get that sum that sum sum for that sum C & E --------------------------------------------------------------------------- 2 1+1 1 3,4,5,6 4 4 3 1+2,2+1 2 4,5,6 3 6 4 1+3,2+2,3+1 3 5,6 2 6 5 1+4,2+3,3+2,4+1 4 6 1 4 --------------------------------------------------------------------------- Total = 20 So the numerator of the probability is 20. Therefore the desired probability is 20 out of 216 or which reduces to Edwin```
 Polynomials-and-rational-expressions/616197: (x^2+x-7)(x+2) multipy and check1 solutions Answer 387558 by EdwinParker(16)   on 2012-05-30 13:06:54 (Show Source): You can put this solution on YOUR website!```(x² + x - 7)(x + 2) x²(x + 2) + x(x + 2) - 7(x + 2) x³ + 2x² + x² + 2x - 7x - 14 x³ + 3x² - 5x - 14 To check, pick an arbitrary value for x (other than 0 or 1). I pick x=3 Substitute x=3 in the original expression: (x² + x - 7)(x + 2) (3² + 3 - 7)(3 + 2) (9 + 3 - 7)(5) (12-7)(5) (5)(5) 25 Now substitute x=3 it in the final answer: x³ + 3x² - 5x - 14 3³ + 3(3)² - 5(3) - 14 27 + 3(9) - 15 - 14 27 + 27 - 15 - 14 54 - 15 - 14 39 - 14 25 Since both answers are the same, we are pretty sure we are right. In some rare cases we could still have made a mistake, but it would be very rare. To be even surer, pick another number and substitute. Edwin```