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# Recent problems solved by 'Theo'

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 Divisibility_and_Prime_Numbers/227267: write a 4 digit number that is divisible by both 6 and 9. Explain how you know without dividing that it is divisible by both 6 and 9. 1 solutions Answer 168929 by Theo(3458)   on 2009-10-15 18:28:38 (Show Source): You can put this solution on YOUR website!multiply 60 * 90 to get 3600 this is divisible by 6 and 9 because I created it using factors that contained 6 and 9 3600 / 6 = 600 3600 / 9 = 400
 Miscellaneous_Word_Problems/227229: Allison has more than 2 but less than 3 hours to spend on her homework. She has work in math, english, and social studies. She plans to spend equal amounts of time studying English and studying social studies, and to spend twice as much time studying math as in studying english. a. What is the minimum number of minutes she can spend on english homework? b. What is the maximum number of minutes she can spend on social studies? c. What is the maximum number of minutes she can devote to math?1 solutions Answer 168927 by Theo(3458)   on 2009-10-15 18:25:43 (Show Source): You can put this solution on YOUR website!let m = number of minutes spent on math. let e = number of minutes spent on english. let s = number of minutes spent on social studies. e = s (spend as much time on english as on social studies) m = 2*e (spend twice much time on math as on english) since e = s, we can say m = 2*e or m = 2*s or m = e+s. they would all mean the same thing. let t = total number of minutes allison can study. 2 hours = 120 minutes 3 hours = 180 minutes 120 < t < 180 (number of minutes allows is more than 2 hours * 60 and less than 3 hours * 60) we want to find the minimum e. we want to find the maximum s. we want to find the maximum m. our constraints are: t = m + s + e 120 < t < 180 s = e m = 2*e = 2*s = s+e we have 2e = m we have s = e we have 120 < t < 180 which means that 120 < e + s + m < 180 since m = 2e and s = e, this equation becomes: 120 < e + e + 2e < 180 which becomes: 120 < 4e < 180 we divide all sides of this equation by 4 to get: 30 < e < 45 the minimum e is > 30 the maximum e is < 45 since m = 2e, then: the minimum m is > 60 the maximum m is < 90 the minimum and maximum s is the same as the minimum and maximum e. let's see how this works. if m and e and s are minimized they would have to be > 60 + 30 + 30 = 120 minutes which is the minimum total time allowed. if m and e and s are maximized they would have to be < 90 + 45 + 45 = 180 minutes which is the maximum amount of time allowed. the questions were: a. What is the minimum number of minutes she can spend on english homework? b. What is the maximum number of minutes she can spend on social studies? c. What is the maximum number of minutes she can devote to math? the answer to a is that she can spend a minimum of 30 minutes on english. the answer to b is that she can spend a maximum of 45 minutes on social studies. the answer to c is that she can spend a maximum of 90 minutes on math. assume 30 minutes on english. this means 30 minutes on social studies. this means 60 minutes on math. total time spent is 120 minutes. assume 45 minutes on social studies. this means 45 minutes on english. this means 90 minutes on math. total time spent is 180 minutes. technically the problem states > 120 minutes and < 180 minutes. minimum of 30 minutes or maximum of 45 minutes for english would violate this becdause then the problem should have been stated as >= 120 minutes and <= 180 minutes. this is picking hairs but mathematically you could not state a minimum of 30 minutes. you would have to state a minimum of more than 30 minutes. likewise for the rest. hopefully this is not an issue but i'm just mentioning it in case it comes up. since there is a fixed relationship between m and e and s, a minimum of any one of them automatically implies a minimum of the others and a maximum of any one of them automatically implies a maximum of the others.
 Signed-numbers/227211: I'm in the 6th grade.My question is:Integer rules when the signs are different you are supposed to subtract,are they talking about the signs in paranthes EX is (+9)+(-8)= the two signs are different so do i just subtract both now,because the signs in parantheses are different 1 solutions Answer 168903 by Theo(3458)   on 2009-10-15 17:41:40 (Show Source): You can put this solution on YOUR website!(+9) + (-8) is the same as (+9) - (+8). You would subtract. the answer is 9 - 8 = 1 It's the operator combined with the sign that determines what you do. The sign is within the parentheses. It tells you whether the number is negative or positive. the operator is outside the parentheses. It tells you whether you will be adding or subtracting. adding a negative number is the same as subtracting a positive number. + (-9) is the same as - (+9) when you combine the operator with the sign, the rules are: ++ means add +- means subtract -+ means subtract -- means add the first sign is the operator. the second sign is the sign of the operand (the number you are working on). check the following website out. It might help clarify it. Adding and Subtracting Positive and Negative Numbers
 Quadratic-relations-and-conic-sections/227187: 3. The graph of y=x^2+2x-8 intersects the x-axis at (1) 2 and -4 (2) -2 and 4 (3) -2 and -4 (4) 2 and 4 Please show steps.1 solutions Answer 168900 by Theo(3458)   on 2009-10-15 17:10:39 (Show Source): You can put this solution on YOUR website!set x^2 + 2x - 8 = 0 this would be the point at which the equations crosses the x-axis. factor x^2 + 2x - 8 to get (x+4) * (x-2) = 0 solving this equation will make both of these terms equal to 0. example: divide both sides of the equation by (x-2) and you get (x+4) = 0 divide both sides of the equation by (x+4) and you get (x-2) = 0 x+4 = 0 results in x = -4 x-2 = 0 results in x = 2 that would be selection (1) a graph of the equation is shown below. as can be seen, the graph of the equation x^2 + 2x - 8 crosses the x-axis at x = 2 and x = -4
 absolute-value/227119: Solve the system |y|>/= 2 |x| I think it might be no solution, but I'm not sure. Can someone please help?1 solutions Answer 168896 by Theo(3458)   on 2009-10-15 16:59:13 (Show Source): You can put this solution on YOUR website!Not sure what your equation is. |y| >= 2 has a solution. The solution is y >= 2 or y <= -2 |x| <= 1 also has a solution. The solution is -1 <= x <= 1 |y| >= 2*|x| also has a solution. The solution is y >= 2*|x| or y <= -2*|x| I could go further but it would get more complicated and you might not need to do so. In order to test the solutions out, you need to put in test cases where the value of y is within limits and the value of y is out of limits to see if the equations hold true or not. Let's take |y| >= 2 Let y = -3, -2, 0, 1, 2, 3 We get |y| equal to 3, 2, 0, 1, 2, 3 |y| >= 2 when y = -3, -2, 2, 3 which is consistent with our requirements that y >= 2 or y <= -2. Let's take |x| <= 1 Let x = -2, -1, 0, 1, 2 Then |x| = 2, 1, 0, 1, 2 |x| <= 1 when x = -1, 0, 1 which is consistent with our requirements that -1 <= x <= 1 Let's take |y| >= 2*|x| Let |x| = 5 This means that 2*|x| = 10 Let's take y = -15, -10, 0, 10, 15 Then |y| = 15, 10, 0, 10, 15 |y| >= 10 when y = -15, -10, 10, 15 which is consistent with out requirements that y <= -10 or y >= 10
 Numeric_Fractions/227125: how do i solve c/6=-21 its says solve this rational expression1 solutions Answer 168867 by Theo(3458)   on 2009-10-15 14:09:44 (Show Source): You can put this solution on YOUR website!equation is c/6 = -21 multiply both sides of equation by 6 to get: c = 6*(-21) c = -126 That's what it looks like to me.
 Geometry_Word_Problems/227118: Where is the center of the circle given by the equation (x + 7)2 + (y - 5)2 = 16? 1 solutions Answer 168864 by Theo(3458)   on 2009-10-15 14:06:08 (Show Source): You can put this solution on YOUR website!This looks like the vertex form of the equation for a circle. The general form of that should be: (x-h)^2 + (y-k)^2 = r^2 where: (h,k) is the (x,y) coordinates of the center of the circle. Since -h = +7, then h must be -7. Since -k = -5, then k must be 5. Center of your circle is at x,y) = (-7,5) To graph your circle, we need to solve for y. Your equation is: (x + 7)^2 + (y - 5)^2 = 16 Subtract (x+7)^2 from both sides to get: (y-5)^2 = 16 - (x+7)^2) Take the square root of both sides of the equation to get: y-5 = +/- Add 5 to both sides of the equation to get: y = 5 +/- graph of your equation looks like.....
 Linear_Equations_And_Systems_Word_Problems/227099: The average hourly wage of employees of a certain company is \$9.83. Assume the variable is normally distributed. If the standard deviation is \$4.58, find the probability that a randomly selected employee earns less than \$5.43.1 solutions Answer 168849 by Theo(3458)   on 2009-10-15 12:32:36 (Show Source): You can put this solution on YOUR website!Average iss \$9.83. Normal distribution. Standard deviation is \$4.58. Probability that a randomly selected employee earns less than \$5.43 is..... .168352 Click on the following hyperlink to go to a tool that helps you calculate these things. It's called a z-table calculator. Z Table Normal Distribution Table Calculator Enter 9.83 into the mean input box. Enter 4.58 into the standard deviation input box. Enter 5.43 in the below input box. Select Below Your answer pops out underneath all the selections in blue. This is the top chart at that website. Bottom chart is for other analyses.
 absolute-value/226401: How do I figure out which is greater than the other? |-6| or |3|1 solutions Answer 168592 by Theo(3458)   on 2009-10-14 08:37:26 (Show Source): You can put this solution on YOUR website!absolute value of a number is always positive. If the number in the absolute value sign is negative, you multiply it by -1 to make it positive. If the number in the absolute value sign is positive, you leave it that way. |-6| = 6 |3| = 3 6 is greater than 3.
 Probability-and-statistics/226402: The probability of surviving a certain transplant operation is .75. If a patient survives the operation, the probability that his or her body will not reject the transplant within a month is .80. What is the probability of survivng both of these stages? A.) .15 B.) .60 C.) .20 D.) .55 E.) .95 I HAVE NO IDEA!! HELP PLEASEE 1 solutions Answer 168590 by Theo(3458)   on 2009-10-14 08:35:21 (Show Source): You can put this solution on YOUR website!I believe it would be .75 * .80 75% of the time he would survive the first stage. 80% of that 75% of the time he would survive the second stage. .8*.75 = .6 which is 60%. Looks like answer B.
 Money_Word_Problems/226393: A buyer of tv sets wishes to maintain a markup of 37 1/2% after all mark-downs are taken. Of 25 sets costing \$150 each, he sells 20 at \$265. How much can he mark-down the remaining 5 sets and still realize his mark-up objective?1 solutions Answer 168589 by Theo(3458)   on 2009-10-14 08:32:26 (Show Source): You can put this solution on YOUR website!25 sets cost \$150 apiece. markup of 37.5% would be 150 + .375*150 = 150 +56.25 = 206.25 he needs to collect 25 * 206.25 to get the 37.5% markup on all the tv sets he sells. 25 * 206.25 means he needs to collect 5156.25 25 * 150 means the total cost to him was 3750 5156.25 / 3750 = 1.375 - 1 = .375 * 100% = 37.5% He sold 20 at \$265 apiece meaning he collected 20 * 265 = 5300. He needs to collect 5156.25 minus 5300 to exactly equal his 37.5% markup. 5156.25 minus 5300 = -143.75 / 5 = -28.75. This means he can give the remaining 5 sets away plus give each person an additional \$28.75 in order to realize his original markup. His cost for the set is \$150. He gives it away plus an addition \$28.75. His markdown is \$150 plus \$28.75 equals \$178.75 / \$150.00 = 1.11916666667 * the original price equals 111.9166666667%. Here's how that would work. Original cost is \$150 * 25 = \$3750. Needs to make (\$150 + .375*150) * 25 = 206.25 * 25 = \$5156.25 Sold 20 at \$265 = \$5300.00 Gave away 5 plus \$28.75 = -\$143.75 Add \$5300 and -\$143.75 to get \$5156.25 Markdown on the remaining 5 is 111.9166666667% He made too much money on the original 20. He can give the remaining sets away and still realize more than he wanted to make in the first place. Formula to use would be: (20 * 265) + (5 * x) = 25 * 150 * 1.375 This becomes: 5300 + 5x = 5156.25 This becomes: 5x = 5156.25 - 5300 = 143.75 This becomes: x = - \$28.75 Markdown is Sale price minus cost, divided by cost. Markdown is therefore -\$28.75 - \$150 = -\$178.75 / \$150 = - -1.19166666667 * 100% equals -111.9166666667 percent. Numbers look a little odd but that's the way it works out.
 Travel_Word_Problems/226394: If a jet travels 5040 kilometers in 6 hours, what is the speed?1 solutions Answer 168582 by Theo(3458)   on 2009-10-14 07:57:52 (Show Source): You can put this solution on YOUR website!rate times time = distance distance = 5040 time = 6 hours rate * 6 = 5040 rate = 5040/6 = 840 kmph (kilometers per hour)
 Geometry_Word_Problems/226384: The area of a circle 8 inches in diameter is greater than the combined areas of a circle 6 inches in diameter and a circle 2 inches in diameter by approximately how many square inches? 1 solutions Answer 168581 by Theo(3458)   on 2009-10-14 07:56:14 (Show Source): You can put this solution on YOUR website!area of a circle is equal to r equals diameter / 2 Circle 1 has a radius of 8/2 = 4 inches. Circle 2 has a radius of 6/2 = 3 inches. Circle 3 has a radius of 2/2 = 1 inch. Area of circle 1 equals = 16* Area of circle 2 equals = 9* Area of circle 3 equals = 1* Differences in the areas is (16-9-1)* equals 6* equals 18.84955592 square inches.
 Pythagorean-theorem/226386: What are the dimensions of a right angle triangle that is similar to the 3 -4 -5 Right-angled triangle, and that has an area four times as large? a.4-5-7 b. 4-12-13 c. 6-8-10 d. 9-12-15 e. 12-16-20 1 solutions Answer 168580 by Theo(3458)   on 2009-10-14 07:49:06 (Show Source): You can put this solution on YOUR website!Area equals bh/2 If you double each dimension, you would probably get an area 4 times as large. Try 6-8-10 base and height will be 3 and 4 in the original triangle to get an area of 12/2 = 6. base and height will be 6 and 8 in the new triangle to get an area of 48/2 = 24 24/6 = 4 meaning the new triangle area is 4 times as large as the original triangle area. Your answer should be C. Your new triangle should also be proportionate to the original triangle. 6-8-10 is exactly double 3-4-5 in the same proportions. Answer B would give you the same area, but the proportions would be off. 3-4-5 and 4-12-13 do not have the same porportions for all of the dimensions. 4/3 not equal to 2 12/4 not equal to 2 13/5 not equal to 2 Answer C keeps the proportions the same 6/3 = 2 8/4 = 2 10/5 = 2 Keeping the sides in porportions means the angles will stay the same meaning you have the same triangle only larger. The triangle would be similar.
 Square-cubic-other-roots/226374: The problem is : (16/9) to the half power ANd this problem is in parenthesis. That is how it was written and it is to the 1/2 power. PLEASE Show me step by step. I have no clue. I need this before class so I hope you know how to show me this. THANKS !! Kristy1 solutions Answer 168579 by Theo(3458)   on 2009-10-14 07:36:30 (Show Source): You can put this solution on YOUR website! is the same as . Since is the same as , then the answer is 4/3 because equals 4 and equals 3. 4/3 = 1.3333333 If you use your calculator to solve , you will get 1.3333333 If you use your calculator to solve , you will also get 1.3333333 If you use your calculator to solve , you will also get 1.3333333 If you use your calculator to solve , you will also get 1.333333 They are all equivalent.
 Inverses/222993: which of the four functions below has an inverse which is the same as the function itself? A: f(x)=x-3 B: f(x)=1/2x C: f(x)=-x-2 D: f(x)= the square root of x+21 solutions Answer 166844 by Theo(3458)   on 2009-10-07 10:59:14 (Show Source): You can put this solution on YOUR website!In all these equations, let y = f(x). To find the inverse function, solve for x and then replace y with x and x with y. alternatively you can replace y with x and x with y first, and then solve for y. Either way you'll get the same answer. selection A y = x+3 Solve for x to get: x = y-3 Replace x with y and y with x to get: y = x-3 which is the inverse function of y = x+3 Selection B y = x/2 Solve for x to get: x = 2y Replace x with y and y with x to get: y = 2x which is the inverse function of y = x/2 Selection C y = -x-2 Solve for x to get: x = -y-2 Replace x with y and y with x to get: y = -x-2 which is the inverse function of y = -x-2 Selection D y= sqrt(x+2) Solve for x as follows: Square both sides to get: y^2 = x+2 Subtract 2 from both sides to get: y^2-2 = x which is the same as: x = y^2-2 Replace x with y and y with x to get: y = x^2-2 which is the inverse function of y = sqrt(x+2) Selection C inverse equation is the same as the original equation. graph of this equation is shown below: The line y = -x-2 is a reflection about the line y = x which is a definition of inverse function. Take any point (x,y) on the line y = -x-2 above the line y = x. The opposite point (y,x) on the line y = -x-2 below the line will be the same distance from the line y = x. example: let x = -2 then y = -(-2)-2 = 2-2 = 0 your coordinate point is (-2,0). now let x = 0 then y = 0-2 = -2 your coordinate point is (0,-2) the point (-2,0) is a reflection of the point (0,-2) about the line y = x to prove that the distance between these points and the line y = x is the same, we need to find the point of intersection between these two lines. the point of intersection with the line y = x would be (-1,-1) as shown on the graph. The distance between the point (-2,0) and (-1,-1) is given by the equation: which equals The distance between the point (0,-2) and (-1,-1) is given by the equation: which equals The point (-2,0) is a reflection of the point (0,-2) about the line y = x which is a requirement of inverse functions. Selection C is your answer.
 Miscellaneous_Word_Problems/222953: At Euclid High School, the number of students taking the AMC10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, and 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? (A) 2002 and 2003 (B) 2003 and 2004 (C) 2004 and 2005 (D) 2005 and 2006 (E) 2006 and 20071 solutions Answer 166834 by Theo(3458)   on 2009-10-07 09:12:31 (Show Source): You can put this solution on YOUR website!60 = 2002 66 = 2003 70 = 2004 76 = 2005 78 = 2006 85 = 2007 rate of increase for each year would be the value in that year minus the value in the previous year divided by the value in the previous year. rate of increase in 2003 over 2002 = 6/60 = .1 rate of increase in 2004 over 2003 = 4/66 = .06 rate of increase in 2005 over 2004 = 6/70 = .085 rate of increase in 2006 over 2005 = 2/76 = .026 rate of increase in 2007 over 2006 = 7/78 = .089 percent equals 1005 * rate. biggest increase percent is 2003 over 2002 = 10%
 Polynomials-and-rational-expressions/222957: Can someone please help me with this. I am studing for an exam and I am sure there will be similar questions. Thanks!! Amount of an investment. The amount of a investment of P dollars are t years at simple interest rate r is given by a=p + prt 1) rewrite this formula by factoring out the GCF on the right side. 2) find a if \$8300 is invested for 3 years at a simple interest rate of 15%. 1 solutions Answer 166828 by Theo(3458)   on 2009-10-07 08:54:59 (Show Source): You can put this solution on YOUR website!The amount of a investment of P dollars are t years at simple interest rate r is given by a=p + prt 1) rewrite this formula by factoring out the GCF on the right side. 2) find a if \$8300 is invested for 3 years at a simple interest rate of 15%. your formula is: a = p + p*r*t where: r is the interest rate t is the number of years you factor out the p on the right hand side of the equation to get: a = p * (1 + r*t) if you invest \$8300 for 3 years at 15% interest per year, then this formula would become: a = 8300 * (1 + (.15*3)) the interest rate of 15% has to be converted to a rate before you do anything with it. rate = percent divided by 100%. 15% / 100% = .15 which is the interest rate per year. your answer would be: a = 8300 * (1 + (.15*3)) this would become: a = 8300 * (1 + .45) this would become: a = 8300 * 1.45 this would result in: a = 12035 15% of 8300 = .15 * 8300 = 1245 * 3 = 3735 + 8300 = 12035 confirming the answer is correct.