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# Recent problems solved by 'bucky'

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 Linear-systems/120711: word problem: Bob climbed down a ladder from his roof, while Rob climbed up another ladder next to him. Each ladder had 30 rungs. Their friend Jill recorded the following info: Bob went down 2 rungs every second, Rob went up 1 rung every second. At some point they were at the same hight, what rung were they on. I know the answer is ten, but I can't come up with the 2 equations that I need to get the answer. We are working on finding points of intersection. Can you help me? Thank!1 solutions Answer 88512 by bucky(2189)   on 2008-01-16 22:56:26 (Show Source): You can put this solution on YOUR website!Let H represent the height of the individuals in rungs and t represent the time in seconds that passes while they climb. . Bob starts at rung 30 and goes down at a rate of -2 rungs per second. So the equation for Bob's height is: . H = -2*t + 30 . Meanwhile, Rob starts at rung 0 and goes up at a rate of +1 rung per second. So the equation for Rob's height is: . H = t + 0 = t . You are interested in finding the height (in rungs) where the two will be standing on the same rung. Since the two heights are the same, we can set the right sides of our two height equations equal. So we can say that: . -2*t + 30 = t . To solve this equation for t, first get rid of the 30 on the left side by subtracting 30 from both sides of the equation. This subtraction converts the equation to: . -2*t = t - 30 . Next get rid of the t on the right side by subtracting t from both sides. The equation then becomes: . -3*t = -30 . Solve for t by dividing both sides of this equation by -3 to find that: . t = -30/-3 = 10 seconds . In 10 seconds Rob will have climbed to rung number 10 because he climbs at 1 rung per second. And in 10 seconds at a rate of 2 rungs per second Bob will climb down 20 rungs. But Bob started on rung 30 so he will be at rung 10 also because 30 - 20 = 10. . Hope this helps you to understand the problem a little better. .
 Angles/120651: Two congruent angles are supplementary. Find the measure of both angles.1 solutions Answer 88506 by bucky(2189)   on 2008-01-16 22:35:25 (Show Source): You can put this solution on YOUR website!Call the two angles A and B . You are told that the two angles are supplementary. That means that when you add the measure of angle A (call it mA) and the measure of angle B (call it mB) the resulting sum is 180 degrees. This relationship can be written in equation form as: . mA + mB = 180 . You are also told that the two angles are congruent. This means that their measures are equal. You can write this relationship as the equation: . mA = mB . From this second equation you can see that wherever you have mA you can substitute mB in its place because they are equals. So go back to the equation: . mA + mB = 180 . In place of mA substitute mB. This makes the equation become: . mB + mB = 180 . On the left side you can see that the sum is 2 times mB or 2*mB. Make this simplification to get: . 2*mB = 180 . To solve for mB divide both sides of this equation by 2. When you do that division the equation reduces to: . mB = 180/2 = 90 . This tells you that the measure of angle B is 90 degrees, and that means that the measure of angle A is also equal to 90 degrees because the two angles are congruent. . Hope this helps you to understand the problem. .
 Graphs/120499: How do I determine which two equations represent parallel lines? (a) y = 5/3x + 4 (b) y = 3/5x - 7 (c) y = 2x + 8 (d) y = 2x - 4??1 solutions Answer 88345 by bucky(2189)   on 2008-01-16 12:26:01 (Show Source): You can put this solution on YOUR website!All of these equations are in the slope-intercept form. This form is: . y = m*x + b . where m ... the multiplier of x ... is the slope and b is the value on the y-axis where the graph crosses the y-axis. . From this you can tell that the slope of the graph for equation (a) is (5/3) because (5/3) is the multiplier of x. . Similarly the slope of (b) is (3/5) . The slope of (c) is 2 . and the slope of (d) is 2 . If the lines are to be parallel, they must have the same slopes. (You might be able to visualize that if the slopes are different, the lines have to intersect at some place.) . Notice that equations (c) and (d) both have the same slope and therefore, they are the pair of graphs that are parallel. . To help you visualize the graphs ... here they are. The graph of (a) is in "red" ... (b) is in green ... (c) is in blue ... and (d) is in purple. You can see that the blue and purple graphs (c) and (d) are parallel. . . Hope this helps you to understand the problem. .
 Linear_Algebra/120518: Hello, can you please help me work out this problem? Find the list price of if there was a 12% discount and the sale price was \$ 4400.00 dollars1 solutions Answer 88336 by bucky(2189)   on 2008-01-16 12:09:52 (Show Source): You can put this solution on YOUR website!The list price is the original price ... that is the price the article was before a 12% discount was taken. . The list price minus the 12% of the list price mark-down results in the \$4400 discounted price. If L represents the list price the equation for the discount is: . L - 0.12L = 4400 . which represents the list price minus 12 percent (0.12) of the list price resulting in 4400 dollars. . Doing the subtraction on the left side reduces the equation to: . 0.88L = 4400 . You can now solve for L by dividing both sides of the equation by 0.88 to get: . L = 4400/0.88 = 5000 . So the original list price was \$5000.00 . Check: . The original list price was \$5000. The 12% discount was 0.12 times \$5000 and this product is \$600. So the discounted price is \$5000 less \$600 which does equal \$4400, just as the problem said it should. . Hope this helps you to understand the problem a little better. .
 Graphs/120501: How do I find the y-intercept for -x + 3y = 15?1 solutions Answer 88333 by bucky(2189)   on 2008-01-16 12:00:02 (Show Source): You can put this solution on YOUR website!Think about a coordinate system. If a point is on the y-axis, what is its corresponding x-value. You can see that any point on the y-axis has zero for its x-value. . Therefore for the given equation, if you set the value of x at zero, the resulting value of y will be the value on the y-axis where the intercept occurs. So ... start with: . -x + 3y = 15 . Set x equal to zero and the equation reduces to: . 3y = 15 . Solve for y by dividing both sides of this equation by 3 to get: . y = 15/3 = 5 . The graph crosses the y-axis where the value of y is +5. You may also see the y intercept given in the form of (x, y). So, from the discussion above, in this form the y-intercept is expressed as (0, +5). . The graph of the given equation looks like this: . . and you can see the y-intercept on this graph is at y = +5 . Hope this helps you to understand the problem a little better. .
 Linear-equations/120415: Algebra: finding temperature Celsius 104F = ? Celsius1 solutions Answer 88293 by bucky(2189)   on 2008-01-16 00:27:19 (Show Source): You can put this solution on YOUR website!The formula for converting a Fahrenheit temperature to a Celsius temperature is: . . If you substitute 104 for F, the equation becomes: . . Inside the parentheses you can subtract 32 from 104 and you get 72. This reduces the equation to: . . Divide the 9 of the denominator into the 72 of the numerator to get 8. The equation then becomes: . . This tells you that 40 degrees Celsius is equivalent to 104 degrees Fahrenheit. . Hope this helps you to understand the process for converting Fahrenheit temperatures to their corresponding Celsius temperature. .
 Equations/120429: Hi! May I have some help with this equation? Evaluate the expression and write the result in the form a + bi. 1/(4 - 7i)1 solutions Answer 88292 by bucky(2189)   on 2008-01-16 00:02:42 (Show Source): You can put this solution on YOUR website!Given: . . You can convert the denominator of this fraction to a real number by multiplying the denominator by its conjugate ... the same term as the denominator only with the opposite sign between the terms. In this case, the conjugate is . If you multiply the denominator by its conjugate, you must also multiply the numerator by the same conjugate. . This multiplication leads to: . . When you multiply the denominators: . . You can do so by multiplying the 4 in the first set of parentheses by both terms in the second set of parentheses and then multiplying the -7i from the first set of parentheses by both terms in the second set of parentheses: . . Notice that the +28i and the -28i are equal but of opposite sign. Therefore, they cancel each other out and you are left with: . . But, by definition, . Substitute -1 for and the expression becomes: . . So the denominator, when multiplied by its conjugate, becomes 65. . The numerator, when multiplied by the conjugate, is: . . This numerator is then over the denominator 65 to give: . . So the answer to this problem, in the form a + bi, is: . . Hope this helps you to understand the problem. .
 Expressions-with-variables/120433: The parabola y=x2-4 intersects the x axis at two points, P and Q. What is the distance from P to Q?1 solutions Answer 88289 by bucky(2189)   on 2008-01-15 23:36:54 (Show Source): You can put this solution on YOUR website!Given: the parabola . . To find the intersections with the x-axis, set y = 0 because any point on the x-axis has zero as its y-value. So the equation to solve then becomes: . . Solve this by getting rid of the -4 on the right side by adding +4 to both sides. On the right side the -4 and the +4 cancel each other when they are added. So when you add +4 to both sides the equation becomes: . . Solve for x by taking the square root of both sides to get two answers: . and . So the intersections of the parabola and the x-axis occur on the x-axis at x = -2 and at x = +2. These points are P and Q. And, if you subtract them you get: . . The points P and Q are 4 units apart. . You can see this in the graph of the parabola shown below: . . Hope this helps you to understand the problem.
 test/120442: the circumference of a circle is 56.52 inches, to the nearest tenth of an inch find the area of the circle. use pi=3.141 solutions Answer 88285 by bucky(2189)   on 2008-01-15 23:21:04 (Show Source): You can put this solution on YOUR website!The equation for the circumference of a circle is: . . Where C represents the circumference and R represents the radius. . The problem tells you that C = 56.52 inches and . Substituting these two values into the equation for the circumference results in: . . Multiply out the constants on the right side to get: . . Solve for R by dividing both sides of the equation by 6.28 to get: . . The radius of the circle is 9 inches. . The area of a circle is given by the formula: . . in which A represents the area, again , and for this problem R = 9 inches. Substituting the values into the equation results in: . . The area of the circle is, therefore, 254.34 square inches. The problem tells you to round this to the nearest tenth of an inch. Rounding gives you the final answer of 254.3 square inches. . So the answer is that the Area of this circle is 254.3 square inches. . Hope this answer helps you to see your way through this problem. .
 test/120443: mr m has 3 times as many quarters as dimes, if he has a total of \$6.80 in quarters and dimes, how many of each does he have. i know this is an easy question i just dont know how to do it algebraically1 solutions Answer 88279 by bucky(2189)   on 2008-01-15 22:58:23 (Show Source): You can put this solution on YOUR website!Let D be the number of dimes and Q be the number of quarters. . Since there are 3 times as many quarters as dimes, if you multiply the number of dimes by 3 the answer will be the same as the number of quarters. In equation form this becomes: . 3D = Q . Each dime is 0.1 of a dollar. So if you multiply the number of dimes D by 0.1 you get the amount in dollars that come from dimes. The resulting term is 0.1D. . Each quarter is 0.25 of a dollar. So if you multiply the number of quarters Q by 0.25 you get the amount in dollars that come from quarters. The resulting term is 0.25Q. . The problem tells you that adding these two terms will result in \$6.80. In equation form this is: . 0.1D + 0.25Q = 6.80 . From our first equation we know that Q = 3D. This tells us that we can go to our dollar equation and substitute 3D for Q. This substitution leads to: . 0.1D + 0.25(3D) = 6.80 . Multiply out the second term on the left side and you get: . 0.1D + 0.75D = 6.80 . Add the two terms on the left side to get: . 0.85D = 6.80 . Solve for D by dividing both sides of this equation by 0.85: . D = 6.80/0.85 = 8 . This tells you that the number of dimes equals 8 . Since the number of quarters is three times the number of dimes, then the number of quarters is 8*3 = 24. . Let's check ... each 4 quarters is a dollar ... and there are six groups of 4 quarters, then the number of dollars from quarters is \$6.00 . and the 8 dimes is 80 cents or \$0.80. . The sum of these two amounts is \$6.00 + \$0.80 = \$6.80, just as the problem said it should. . And with 24 quarters and 8 dimes, there are 3 times as many quarters as there are dimes. This is also as the problem said it should be. Therefore, the answer of 24 quarters and 8 dimes fully checks out. . Hope this helps you to understand the problem and how to solve it algebraically. .
 Systems-of-equations/120448: Three times the larger of two numbers is equal to four times the smaller. The sum of the numbers is 21. Find the numbers1 solutions Answer 88274 by bucky(2189)   on 2008-01-15 22:29:12 (Show Source): You can put this solution on YOUR website!The first sentence of the problem indicates that you are looking for two numbers. It also tells you that there is a larger number (let's call it L) and as smaller number (let's call it S). Finally, it tells you that 3 times the larger (3*L) is equal to 4 times the smaller (4*S). In equation form this can be written as: . 3*L = 4*S . The sum of the numbers is 21. Therefore, another equation is: . L + S = 21 . Let's solve these two equations by substitution. We can start by solving the second equation for L by subtracting S from both sides to leave just L by itself on the left side. When you subtract S from both sides, the equation becomes: . L = 21 - S . Now let's substitute the right side of this equation for L in the first equation that we wrote. The first equation is: . 3*L = 4*S . substituting 21 - S for L changes the equation to: . 3*(21 - S) = 4*S . Do the distributed multiplication on the left side by multiplying 3 times each of the terms in the parentheses. 3 times 21 is 63 and 3 times -S is - 3*S. This changes the left side and the equation becomes: . 63 - 3*S = 4*S . Get rid of the -3*S on the left side by adding +3*S to both sides. On the left side, this addition cancels the -3*S and on the right side the 3*S and 4*S add to give 7*S. So the changed equation is: . 63 = 7*S . Solve for S by dividing both sides of the equation by 7. This makes the answer for S: . S = 63/7 = 9 . Now you know that the smaller number is 9. And since the total of the two numbers is 21, the larger number is 21 - 9 = 12. . Check: 3 times the larger number is 3*12 = 36. And 4 times the smaller number is 4*9 = 36. This means that 3*L = 4*S . And L + S is 12 + 9 and that does equal 21, just as it should. . Therefore, the answers of 12 and 9 satisfy the problem and are, therefore, the correct answers. . Hope this helps you to understand the problem. .
 test/120359: the diameter on a wheel on a car is 1.5 feet. how many revolutions must the wheel make so that it travels 37.68 feet. use pi=3.14 1 solutions Answer 88225 by bucky(2189)   on 2008-01-15 18:13:02 (Show Source): You can put this solution on YOUR website!The circumference of the wheel is found using the formula: . C = pi*D . where C is the circumference and D is the diameter. So for this problem: . C = pi*1.5 = 3.14*1.5 = 4.71 feet . Each turn of the wheel moves the car forward a distance equal to the circumference of the wheel . To move the car forward 37.68 feet the number of rotations (revolutions) the wheel must make is found by dividing 4.71 into 37.68 .... . Rotations = 37.68/4.71 = 8 . So, if the wheel rotates 8 times, the car will move forward 37.68 feet. . Hope this helps you to understand the problem. .
 Quadratic_Equations/120256: Using the following sequence (-1,-5,-9,-13,...) find the function tn.1 solutions Answer 88125 by bucky(2189)   on 2008-01-15 08:18:19 (Show Source): You can put this solution on YOUR website!Given the sequence (-1, -5, -9, -13, ...) . To find the function, first note that the difference between terms is -4 which has to be added to the preceding term to get the next term in the series. . The first term is -1. So you can say that it equals 3 + (-4*1). The second term is then equal to 3 + (-4*2) and the third term is 3 + (-4*3) and so on. . Therefore, to calculate you can use the relationship: . . where n represents the sequence number of the term you are looking for in the series. For example, to find the 6th term in the series, substitute 6 for n and you have: . . Hope this helps you to understand the problem and shows you a thought process that will lead you to the answer. .
 Finance/120132: answer for one video club charges \$25 to become a member and \$2.50 to rent each video another charges no rental fee and \$3.25 to rent each video how many videos must you rent to make the first club more economical1 solutions Answer 88120 by bucky(2189)   on 2008-01-15 08:03:54 (Show Source): You can put this solution on YOUR website!Let C represent the total cost of renting x number of videos. . For the video club that charges a \$25 fee to join, the cost will be that \$25 plus an additional charge of \$2.50 times the number of videos rented (x videos). So the total cost of renting can be written in equation form as: . C = 2.50*x + 25 . The other video club just charges \$3.50 for each video that is rented. So the total cost to rent x videos is just \$3.50 times x. In equation form this is: . C = 3.50*x . The "break-even" point occurs when the two costs are equal. If the two costs are equal, then the right sides of the two equations are equal. Set them equal and you have: . 2.50*x + 25 = 3.50*x . You can solve this equation by first collecting the x-terms on one side of the equation. To do this, get rid of the 2.50*x on the left side by subtracting 2.50*x from both sides of the equation. When you do that subtraction you end up with: . 25 = 1.0*x or just 25 = x . This tells you that when you rent 25 videos the cost is the same for each club. After that, the cost will be less for the club with the \$25 fee, because each video rental beyond 25 rentals will cost \$2.50 ... but each video beyond 25 in the other club will always cost \$3.50 no matter how many you rent. . You can also see this if you graph the two equations. The graph below has the rental cost on the y-axis and the number of rentals along the x-axis. The "red" graph shows the cost of renting tapes from the club that has the \$25 fee. The green graph shows the cost of renting tapes from the club that charges \$3.50 for each rental. . . Note that out to 25 on the x axis, the green graph is below the red graph. That means the total cost of getting up to 25 rentals will be lower than the cost of renting from the club with the \$25 fee. For more than 25 rentals (on the x-axis) the total cost of the red graph will be lower than the total cost of the green graph. Therefore, the cost will be less for the club with the \$25 fee (shown by the red graph being lower than the green graph). . Hope this helps you to understand the problem. .
 Graphs/120231: how do you solve/graph 2x - 7 = 01 solutions Answer 88113 by bucky(2189)   on 2008-01-15 07:34:55 (Show Source): You can put this solution on YOUR website!Given: . . To solve this equation for x, first get rid of the -7 on the left side. You can do that by adding +7 to both sides. Adding +7 to the left side cancels the -7 on the left side and the equation becomes: . . Now solve for x by dividing both sides by 2 and you have: . . This means that x is always 3.5 ... so any point that has 3.5 for its x value will satisfy the equation you were originally given. . As examples, all of the following points would satisfy the original equation: . (3.5, 10) (3.5, 0) (3.5, -15) . because each of those points has an x value of 3.5 . How would the graph look on a coordinate system? . If you plotted the three points above (and any other point that has 3.5 as its value for x) the points would line up vertically through the point +3.5 on the x-axis, and the graph would look like: . . Hope this helps you to understand the problem.