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# Recent problems solved by 'bucky'

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 Miscellaneous_Word_Problems/553719: The first side of a triangle is y yards long. The second side is 3 yards shorter than the first side. The third side is three times as long as the second side. What is the perimeter of the triangle in feet? This is how I evaluated it, but not sure if this equation is correct: y + (y-3) + 3(y -3) / 31 solutions Answer 361011 by bucky(2189)   on 2012-01-07 08:38:58 (Show Source): You can put this solution on YOUR website!I'm not sure why you included the division by 3. It is not needed. . You were correct with: . y + (y-3) + 3(y -3) . To write the equation you can let P represent the perimeter. Then the equation is: . P = y + (y-3) + 3(y -3) . To simplify the right side you can do the distributed multiplication of the last term by multiplying +3 times each of the terms in the parentheses to get: . P = y + (y-3) + 3y - 9 . Because the remaining set of parentheses (for the second side) is preceded by a + sign, you can just remove the parentheses and you then have: . P = y + y - 3 + 3y - 9 . Now add up the three terms that contain y ... y + y + 3y = 5y . and you have P = 5y - 3 - 9 . Finally combine the two constants -3 - 9 = -12 . and you have the final equation: . P = 5y - 12 . and don't forget that the units for P is yards. . Hope this helps you to understand the problem better. You had done the hard part correctly except that there was no need to divide the last term by 3. The third side was just three times the second side, and since the second side was y - 3, you only needed to multiply 3(y - 3) to get the third side. . Keep up the good work ... you're getting it!!!
 Graphs/551233: how do you solve for the unknown of x/3 -7/8 = 2/5(x + 15/4) any help would be greatly appreciated.1 solutions Answer 359560 by bucky(2189)   on 2011-12-31 11:48:10 (Show Source): You can put this solution on YOUR website!Given to solve for x: . . There are a number of different things we could do to make the problem a little easier. A lot of students have trouble with fractions, so let's begin by getting rid of the fractions. The right side of the equation may cause us a little trouble, so let's begin by doing the distributed multiplication on the right side. Do that by multiplying times each of the two terms in the parentheses to get: . . When you multiply out the last term on the right side [that is when you multiply ] you get which is . So you can now write the equation as: . . Now to get rid of the fractions, we can multiply every term on both sides by the product of all the denominators. That is multiply every term by . (We could use 120, but using 240 does not cause a problem). . When we multiply by 240 the equation becomes: . . Now in each of the four terms, let's divide the denominator into the 240 in the numerator to get: . . Multiply out all the terms and you have: . . This is in a form that is more like an ordinary algebraic equation. Start by subtracting 96x from both sides: . . Next add 210 to both sides: . . Solve for x by dividing both sides by -16 to get: . . and since both 570 and -16 are divisible by 2 this can be reduced to: . . Or by dividing it out, you can get the decimal equivalent form: . . Hope this helps you to understand the problem a little better. Sometimes it helps to make the problem a little easier by getting rid of the fractions, and you can do that by multiplying all terms by a common denominator (the product of all denominators) and then getting rid of each denominator by dividing it into the common denominator. .
 Numbers_Word_Problems/550866: Greg's family rode in a taxicab last satuday night. The total cost of the ride, with a 20% tip, was \$25.80. What was the cost of the ride without tip?1 solutions Answer 359155 by bucky(2189)   on 2011-12-29 13:28:38 (Show Source): You can put this solution on YOUR website!Let C represent the cost of the ride without a tip. So when Greg's family arrived at their destination, the meter said that they owed the driver C. . They apparently got good service so they decided to add a 20% tip. That means that they multiplied the cost on the meter by 0.2 (0.2 is equivalent to 20%). and they added that to C. When they did that the total was \$25.80. In equation form you can write that: . C + 0.2*C = \$25.80 . You can add the two terms on the left side because they both involve C. The 1*C plus 0.2*C equals 1.2*C and this makes the equation become: . 1.2*C = \$25.80 . Now you can solve for C by dividing both sides of the equation by 1.2 to get: . C = \$25.80/1.2 = \$21.50 . So you now know that C (the cost of the ride) was \$21.50. To that amount they added a 20% tip (0.2 times \$21.50 = \$4.30). So the \$21.50 for the ride and the \$4.30 for the tip add up to \$21.50 + \$4.30 = \$25.80, just as the problem said it should. . In summary, the cost of the taxicab ride was \$21.50 . Hope this helps you to understand the problem a little better. .
 Triangles/550848: what is area of triangle with base is 100ft and height is 70ft1 solutions Answer 359150 by bucky(2189)   on 2011-12-29 12:46:33 (Show Source): You can put this solution on YOUR website!The formula for the area (A) of any triangle is: . . in which B represents the dimension of the Base and H represents the dimension of the Height or Altitude. . For this problem you are given that B = 100 ft and H = 70 ft. Substituting these two values into the formula results in: . . The 1/2 times 100 results in 50 and then multiplying the 50 times 70 results in 3500. The units are square feet. So the answer you are looking for is 3500 square feet. . Hope this helps you to understand this problem. .
 Travel_Word_Problems/550625: Suppose the tip of the minute hand of a clock is two inches from the center of the clock. For each of the following durations, determine the distance traveled by the tip of the minute hand. Question 1: 15 minutes and Question 2: 10 and a half hours. Not sure at all how to approach this problem. 1 solutions Answer 358925 by bucky(2189)   on 2011-12-28 10:51:45 (Show Source): You can put this solution on YOUR website!The minute hand will make one complete rotation in an hour (60 minutes). Picture that at the start of an hour, the minute hand is pointed at the 12 on the clock. It then begins a slow rotation around the clock until 60 minutes later it arrives at the position where it again points to the 12. How far has the tip of the minute hand traveled during that hour? It has traveled the complete circumference of a circle that has a radius of 2 inches. So you can write an equation for this distance as: . . Where C stands for the circumference of the circle and r stands for the radius, and the equation is just the standard equation for finding the circumference of a circle if you know the radius. . In this problem, the radius is 2 inches and therefore you can substitute 2 for r and you will find the circumference in inches. This is as follows: . . which by multiplying out the right side simplifies to: . inches . You can convert this to a more common form by substituting 3.1416 as the value for and multiplies out as follows: . and this becomes inches per hour. . So for each hour the tip of the minute hand travels inches or an equivalent numerical value of 12.5664 inches. . Now to answer the questions that you are asked in the problem. First, how far does the tip of the minute hand travel in 15 minutes? 15 minutes is a quarter (one-fourth) of an hour. So in 15 minutes the tip of the minute hand travels one-fourth of the distance it would travel in an hour. So, using D to represent the distance the tip moves we can say that for 15 minutes the tip moves: . . And dividing out the right side, you get that in 15 minutes the tip of the minute hand moves: . inches which numerically is inches . Next, how far does the tip of the minute hand travel in 10 and a half hours? Since each hour it travels inches, in 10 hours it will travel a distance of: . which multiplies out to inches. . Then in the additional half hour the tip, since the tip moves inches in an hour, it will move half that distance in the half hour. In other words, in the half hour it moves half of inches, or it moves inches. So in 10 and a half hours it moves the sum of these two distances which is: . inches and this totals to inches . As before you can convert this to numerical inches by substituting 3.1416 for to get: . inches and this multiplies out to inches. . Hope this helps you to understand how to work clock problems such as this one. .
 Graphs/550022: Find an equation in slope-intercept form passing through the points (-6.2,18.1) and (-3.4,18.1).1 solutions Answer 358328 by bucky(2189)   on 2011-12-23 18:33:43 (Show Source): You can put this solution on YOUR website!This one might be a little confusing to you. . Normally, the slope intercept form is: . . in which m, the multiplier of x, is the slope and b is the value on the y-axis where the linear graph of the equation crosses the y-axis. . The slope is normally computed by dividing the change in x from the two given points into the corresponding change in y. In this case, note that in the two points that are given, x goes from -6.2 in one of the points to - 3.4 in the other. Subtracting the -6.2 from -3.4 results in a difference of +2.8. So in finding the slope, +2.8 is what we'll be dividing into the corresponding change in the y values of the two points. . But, what is the change in y? The value of y in the first point is 18.1 and the value of y in the second point is also 18.1. The difference in these two values is zero. So while x went 2.8 units toward the right (the run from -6.1 to -3.4) the value of y stayed the same and, therefore, the rise was zero. Dividing 2.8 into zero results in a slope of zero. . Return to the the slope-intercept form for the equation. When you substitute zero for the slope the equation becomes: . . The multiplication by zero makes the x term disappear and we are left with: . . Now all we have to realize is that y never changes regardless of what value we choose for x. For this problem the value of y is always 18.1 and the graph is a horizontal line (parallel to the x-axis) that crosses the y-axis at +18.1. This means that b is 18.1 and the slope intercept equation is simply: . . That's all there is to it. Whenever you see a slope intercept form that says simply y equals a constant, you should picture a graph that is a horizontal line and crosses the y-axis at the value of the constant. . Hope this helps you to understand the "trick" in this problem and what it means. .
 absolute-value/549539: find the solution of the following x-5>01 solutions Answer 357838 by bucky(2189)   on 2011-12-21 09:50:47 (Show Source): You can put this solution on YOUR website!You can work these problems using the same rules that you have learned for solving equations with this exception: If you multiply or divide both sides of the inequality BY A NEGATIVE Quantity, you must reverse the direction of the inequality sign. . So, just imagine for the time being that the > sign in this problem is an equal sign. . You are given: . . You are expected to solve this for x. In an ordinary equation you would get rid of the -5 on the left side by adding 5 to both sides. You do the same here. Just add 5 to both sides and the problem becomes: . . And that's the answer to this problem. x is greater than 5. . We didn't make use of the exception because we didn't have to multiply or divide both sides by a negative quantity to solve for x. But don't forget that rule. It will be useful for other problems of this type. . For the time being just think of working these problems by following the rules you already know for equations. . Hope this helps you to understand inequalities a little better.
 logarithm/549501: 2log(x)-log(10)-3=01 solutions Answer 357835 by bucky(2189)   on 2011-12-21 09:35:43 (Show Source): You can put this solution on YOUR website!Given to solve: . I think I'm safe in assuming that by using the term "log", you mean that the base of the logarithms you are using is 10. . One of the first things you can choose to do is to add +3 to both sides of this equation to get rid of the -3 on the left side and move the constant to the right side as follows: . . Now you have only logarithm terms on the left side. You can apply the rules of logarithms to these terms. Begin by noting that the first logarithm term has the constant 2 multiplying it. By the rules, a multiplying constant can be taken inside the logarithm function as an exponent as shown below: . . There are now two ways that you can deal with the negative logarithm. You can note that when the base of the logarithm is the same as the quantity that the logarithm is operating on, the logarithm term can be replaced by the number 1. Here are some examples: . ; ; ; and . So you could immediately replace by and you would have: . . Then you could add 1 to both sides to get rid of the -1 on the left side. The result would be: . . And you could proceed to solve for x from this point by using the conversion from logarithmic form to exponential form that I'm going to describe a little bit further on. . But instead of doing it that way let's do it another way that contains the lesson that the difference between two logs can be expressed as the logarithm of their division in which the denominator is the quantity that the negative logarithm term is operating on. Sounds hard, but it's pretty easy actually. Going back to the point we had arrived at, which was: . . The rules of logarithms say that you can convert the two terms on the left to a single logarithm in which the quantity from the first logarithm is divided by the quantity from the negative logarithm as follows: . . Next we're going to take a look at converting logarithms to exponential form. This is an important property to learn because so many logarithm problems (like this problem) make use of it. The conversion equation says that by the definition of logarithm: . is the same as saying . This says "If you have a logarithm form, you can change it to an exponential form by taking the base of the logarithm, raising it to the exponent of the quantity on the other side of the equal sign, and setting that result equal to the quantity that the logarithm is operating on." Let's continue. We have arrived at: . . Let's convert this to exponential form. Take the base (which is 10); raise it to the power of the 3 on the right side of this equation, and set that equal to . This leads to the exponential equation: . . I hope that you followed that. Now get rid of the denominator 10 by multiplying both sides of this equation by 10. On the left side multiplying . . and by adding the exponent 3 to the exponent 1 implied for just the 10) this multiplication results in: . . and on the right side, the multiplication by 10 eliminates the 10 in the denominator. So this equation has been reduced to: . . Now you can solve for x simply by taking the square root of both sides. When you do that the exponent on the left side is just divided by 2 to give you the square root (think . And on the right side the x-squared becomes just x. So we have the solution: . . And the 10 squared is just 10 times 10 which is 100. . So the answer to this problem is x = 100. . Hope that this long explanation helps you to understand logarithms a little better. It's just a matter of practice and thinking about all this. It'll become easier with working on these types of problems until you get familiar with all the rules of logarithms. Keep working at it. .
 logarithm/549397: I'm completely done with my assignment, But i'm on my last question and it's a critical thinking question. It says, 2(log 2x - log y)-(log 3 + 2log 5). I got told the answer is: log 4x^2/75y^2 but i don't know how to work it out and explain my work. It's a difficult question and if you could help me that would be great. Thanks for your help. 1 solutions Answer 357717 by bucky(2189)   on 2011-12-20 18:01:21 (Show Source): You can put this solution on YOUR website!Given to simplify: . . Look at the first term, specifically look inside the parentheses. The parentheses contain the difference of two logarithms and by the rules governing logarithms, the difference of the logarithms of two quantities is equal to the logarithm of the division as shown in the following: . . Next by the rules of logarithms, the 2 that multiplies the logarithm of the simplified first term can become the exponent of the quantity that the logarithm is operating on. When this rule is applied, the first term changes as shown in the following expression: . . Still operating on the first term, use the power rule of exponents to square the quantity that the log function is operating on and get: . . Moving on to the second term in the expression, change the 2 that multiplies log(5) so that it becomes the exponent of 5 as in the following: . . and square the 5 to get: . . Now recognize that the sum of two logarithms equals the log of the product of the quantities that the two logarithms are operating on: . . Multiply out the 3*25 to get 75 so that the expression now is: . . You again have the difference of two logarithms which means that by the rules of logarithms you can change this to the logarithm of the division of the two quantities that the logarithms are operating on: . . Dividing by 75 is the same as multiplying by 1/75 and this gives you the final simplified answer of: . . Hope this helps you to understand the problem. Make sure you understand each of the rules for logarithms that are applied in solving this problem. .
 Inequalities/549260: x + 2 < √5 - x1 solutions Answer 357567 by bucky(2189)   on 2011-12-20 01:00:23 (Show Source): You can put this solution on YOUR website!Given to solve: . . Add x to both sides to eliminate the -x on the right side. When you do that, the inequality becomes: . . Subtract 2 from both sides to eliminate the +2 on the left side. You get: . . Divide both sides by 2 to solve for the limits on x: . . And that's the answer. . Note that you can work these inequalities just as you would an equation, with the exception that if you divide or multiply by a negative quantity you must reverse the direction of the inequality sign. This exception did not apply in this problem that you were given to solve. . Hope this helps you to better understand how to work inequality problems. .
 Linear-systems/548732: what is the slope of the following line? 3x+7y=41 solutions Answer 357189 by bucky(2189)   on 2011-12-18 15:55:05 (Show Source): You can put this solution on YOUR website!Given the equation: . You are to find the slope of the graph for this equation. . One way of doing this is to rearrange the given equation so it is in the slope-intercept form which is: . . When you get it into this form, m, the quantity multiplying the x term, will be the slope, and b will be the value on the y-axis where the graph crosses. . Start with: . and the plan is to solve for y with everything else on the right side. . Start by subtracting 3x from both sides. When you do that the 3x disappears on the left side and the equation becomes: . . Next solve for y by dividing both sides (all terms) by 7 to get: . . This is in the slope intercept form. You can see that the multiplier of the x term is and that is the value of the slope. The b term is and that is the value on the y-axis where the graph crosses. . So, the slope is and that is the answer to this problem. . Hope this helps you to understand this problem. .
 Equations/548697: Can you please show the work on this for me? x=2 y=-4 the answer is -80 2x^2+3xy-4y^2 I can't get to -80 and not sure how to do the 3xy part? Thank you!1 solutions Answer 357182 by bucky(2189)   on 2011-12-18 15:40:20 (Show Source): You can put this solution on YOUR website!You are asked to evaluate . . when x = 2 and y = -4 . Start by evaluating . When x = 2 then . This is equal to . Next evaluate . Substitute 2 for x and -4 for y to get: . . The 3*2 = 6 and then the 6*-4 = -24 . Finally, evaluate when y = -4. Substitute -4 for y to get: . and this becomes . Now combine all three terms. Don't forget that this last one has a minus sign preceding it. . . Remove the parentheses and then sum the terms on the right side to get the answer: . . Combine the +8 and -24 to get -16. Then add the -16 to the -64 to get -80. . Hope this helps you to find where you were having trouble. In this case it could easily involve being confused by the signs. .
 Linear-equations/548646: Write an equation of the line that passes through the point (7, -6) and is perpendicular to the line 4x + 6y = 71 solutions Answer 357141 by bucky(2189)   on 2011-12-18 12:58:48 (Show Source): You can put this solution on YOUR website!We were given the equation: . . First, we need to find the slope of the graphed line that represents this equation. To do that, let's transform the given equation into the slope-intercept form . Subtract 4x from both sides of the given equation. The result is: . . Solve for y by dividing both sides (all terms) by 6 . . Note that the fraction (4/6) reduces to (2/3) when both the numerator and denominator are divided by 2. This makes the slope-intercept form become: . . Here are the two critical things to this problem: . First, in the slope-intercept form of an equation, the multiplier of the x (called the coefficient of x) is the slope of the graphed line for that equation. . And second, any line perpendicular to the graphed line has a slope that is the negative and inverse of the slope for the graphed line. . So looking at the equation that we have in the slope-intercept form, we see that the slope for its graph is because that is the multiplier of the x. (This -2/3 slope means that for every 3 units you move horizontally to the right in the x direction, the graph goes down 2 units in the y direction.) . Next we know that a line that is perpendicular to it will have a slope that is the negative of this (means change in sign) and is the inverse in value. (An inverse of an integer, means just put that integer as a denominator under a numerator of 1. So, for example, the inverse of 5 is . For fractions, such as we have, we can quickly find the inverse by flipping the fraction upside down.) Applying these two characteristics, we see that the inverse of is and then taking the negative of that changes it to positive. So we found that the slope of the line perpendicular to our given equation is and it is the negative inverse of the slope of the given equation. . Let's put that into a slope-intercept form for the new equation we are getting for the perpendicular. . Again the basic slope intercept form is: . . In which m (the multiplier of x) is the slope we want and b is the point where the graph for this equation crosses or intercepts the y-axis. So let's substitute the slope that we want into this equation to make it: . . Next we make use of the fact that we want the graph of this equation to go through the given point (7, -6). . We know that the x value 7 and its corresponding y value -6 have to work in our equation, meaning that they have to be on the graphed line for this new equation. So we start with: . . and substitute 7 for x and -6 for y to get: . . Multiply out the fraction on the right side to get: . . Solve for b by subtracting 21/2 from both sides as follows: . . and is the same as so we can substitute that to get: . . and combining the two fractions on the left side to get: . . Now we can return to the equation we are finding: . . and substitute -33/2 for b to get: . . and that's the answer: the equation of the answer in slope intercept form for the perpendicular to the graph of the given equation. . We can convert this answer to the same form of the equation that we were given in the original problem. Begin by subtracting from both sides to get: . . Then you can get rid of the denominator 2 by multiplying all term on both sides of this equation by 2 to get: . . The graph of these two equations, the original equation and the equation for the perpendicular to it that goes through (7, -6) is shown below: . . The red graph is for the original equation we were given, and the green graph is the graph of the equation that we developed for its perpendicular. As a check, you should be able to see that the point (7, -6) is on the green graph of the perpendicular. . Hope this helps you to understand how to find perpendiculars to the graphs of equations that you are given. .
 Probability-and-statistics/547845: Two dice are tossed and various amounts are paid according to the outcome. In a certain game if a nine or ten occurs on the first roll the players wins. what is the probability of winning on the first roll.1 solutions Answer 357094 by bucky(2189)   on 2011-12-18 07:48:06 (Show Source): You can put this solution on YOUR website!There are 36 combinations (six outcomes on the first die times 6 outcomes on the second die) that can be rolled when you roll a two dice. . How many different combinations that total to 9 can be rolled? (First die 6, second die 3); (First die 3, second die 6); (First die 5, second die 4); (First die 4, second die 5). So there are 4 possible combinations totaling 9. . How many different combinations that total to 10 can be rolled? (First die 6, second die 4); (First die 4, second die 6); (Both dies 5). So there are 3 possible combinations totaling 9. . This means that there are a total of 7 chances of winning (4 from a 9 and 3 from a 10) by scoring a 9 or a 10 on the first roll. . Therefore, the chance of winning on the first roll is 7 in 36 or the probability is 7 divided by 36 and this is 0.1944 which is expressed as a probability of 19.44%. . Hope this helps you to understand the problem. And to help you with future problems involving a dice pairs, here are the possibilities for each roll: . Rolling a 2: one chance (1&1) Rolling a 3: two chances (2&1)(1&2) Rolling a 4: three chances (3&1)(1&3)(2&2) Rolling a 5: four chances (4&1)(1&4)(3&2)(2&3) Rolling a 6: five chances (5&1)(1&5)(4&2)(2&4)(2&2) Rolling a 7: six chances (6&1)(1&6)(5&2)(2&5)(4&3)(3&4) Rolling an 8: five chances (6&2)(2&5)(5&3)(3&5)(4&4) Rolling a 9: four chances (6&3)(3&6)(5&4)(4&5) Rolling a 10: three chances (6&4)(4&6)(5&5) Rolling an 11: two chances (6&5)(5&6) Rolling a 12: one chance (6,6) . The sum of the numbers of possible outcomes is: 1+2+3+4+5+6+5+4+3+2+1 = 36