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# Recent problems solved by 'bucky'

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 Exponents-negative-and-fractional/69787: How do you solve: x^-2=91 solutions Answer 49752 by bucky(2189)   on 2007-02-08 07:13:27 (Show Source): You can put this solution on YOUR website!Your problem is: The negative exponent means that the term raised to that negative value can be changed into a fraction having 1 as the numerator and the term with a positive exponent as the denominator. Easier to show by an example than to explain it. Think "I can change into by making the denominator of a fraction that has 1 as its numerator." When you do that your problem becomes: Then you can eliminate the denominator if you multiply both sides by If you do that multiplication on the left side you have a fraction that has in both the numerator and denominator. This is equivalent to 1. On the right side the multiplication results in and the problem becomes: Dividing both sides by 9 gives: Now just take the square root of both sides. Notice that because . Also notice that both the positive and negative forms of are needed because when either is squared they produce a positive value of The two answers to this problem are: and You can check this by putting each of these answers (positive and negative) one at a time into the equation: and see that with these values the left side of the equation equals 9.
 Rational-functions/69673: x-7 is in square sign 1 solutions Answer 49634 by bucky(2189)   on 2007-02-07 09:28:04 (Show Source): You can put this solution on YOUR website!If I understand your problem correctly it is: One of the ways to solve this is to take the square root of both sides to get: and Then add + 7 to both sides and the result is: If you calculate this you will find the two answers are about 9.645751311 and 4.354248689 Another way to solve it is to square out the left side of the equation by multiplying . If you do that, the equation becomes: Subtract 7 from both sides to make the equation: This is in the form for applying the quadratic formula that says if a quadractic equation is in the form: then the solution for x is in the form: For this problem a = 1, b = -14, and c = 42 Substituting the values of a, b, and c for this equation results in: If you work this out you will get the same answers as you got previously Hope this helps.
 Linear-equations/69668: Can you help me ith graphing the equation y=3.5?1 solutions Answer 49626 by bucky(2189)   on 2007-02-07 08:44:56 (Show Source): You can put this solution on YOUR website!Think about this ... x can be any value and y will always equal 3.5. If x is 0, then y = 3.5 If x is 1, then y = 3.5 If x is 5, then y = 3.5 If x is -4, then y = 3.5 If x is -10, then y = 3.5 If you plot all these points on a graph you should begin to see that the graph is a horizontal line that intercepts the y-axis at +3.5 and continues on forever in both plus and minus directions. Any time that there is no x term in the equation, the graph is a horizontal line that crosses the y-axis at the given value of y. For example, the graph of y= -4 would be a horizontal line that crosses the y-axis at the point -4 and continues on forever in both directions (to the left and to the right).
 Polynomials-and-rational-expressions/69677: (x/5x+10)+((x-3)/(x+2))=7/5 Solve for x 1 solutions Answer 49619 by bucky(2189)   on 2007-02-07 08:30:21 (Show Source): You can put this solution on YOUR website! You are given: Notice that 5x + 10 can be factored by removing a 5 to make it 5(x+2). If you do this the equation becomes: Next you need to make all the denominators the same. The common denominator is 5(x+2) so you don't need to do anything to the first term. It already has the common denominator. The second term lacks a 5 in the denominator. Therefore, multiply this term by . This is equivalent to multiplying the second term by 1 because is 1. When you do this the equation becomes: Finally, on the right side of the equation the denominator lacks the . Therefore, multiply the right side by [effectively multiplying it by 1. When you do that the equation becomes: Now you can get rid of the common denominator entirely by multiplying all the terms in the equation by . This cancels out the denominator and leaves you with the equation: Its all downhill from here. Just do the distributive multiplication on both sides to get: Add the two terms containing x on the left side to get: Add 15 to both sides: Subtract 7x from both sides: Multiply both sides by -1: If you substitute this value for x in the original equation, you should find that it makes the left side of the equation equal to Hope this helps.
 Linear-equations/69685: Tell the answer to, or how to put, 4x-5y=10 -into slope intercept form Tell the answer to, or how to put, y=-3/2x+4 -into standard form1 solutions Answer 49617 by bucky(2189)   on 2007-02-07 08:00:32 (Show Source): You can put this solution on YOUR website!Tell the answer to, or how to put, y=-3/2x+4 -into standard form The slope-intercept form is in which m and b are constants, in which m is the slope of the graph and b is the value where the graph intercepts the y-axis. So what you need to do for the first problem is to move the terms around and solve for y to get the equation to match the slope intercept form. Move the 4x term to the right side of the equation, by subtracting 4x from both sides. This subtraction cancels out the 4x on the left side and makes a -4x appear on the right side as follows: Now divide all terms on both sides of the equal sign by -5 to get: When you divide -4 by -5 the answer becomes positive. And dividing +10 by -5 results in -2. So the answer becomes: The next problem involves putting an equation into standard form. The standard form is: where a, b, and c are all constants. So what you need to do is take the equation you are given, an equation in slope-intercept form, and rearrange things to get it into standard form. You are given: Get rid of the denominator of 2 by multiplying all the terms in this equation by 2 to get: Next, move the -3x term to the other side of the equal sign by adding 3x to both sides. Adding +3x to the right side cancels the -3x. And adding 3x on the left side makes a +3x appear on the left side. The result is: This is the standard form that you are looking for. Hope this helps.
 Polynomials-and-rational-expressions/69626: Factor by grouping. 6x^3 + 21x^2 – 2x – 7 I came up with (2x+7)(3x^2-1) I hope that someone can tell me if this is correct???? Thank you.1 solutions Answer 49601 by bucky(2189)   on 2007-02-06 22:06:58 (Show Source): You can put this solution on YOUR website!Let's check your answer by multiplying your factors together using the FOIL method (first terms, outside terms, inside terms, and last terms) Your answer was: Multiplying the first terms in each factor results in: Multiplying the outside terms in each factor results in: Multiplying the inside terms in each factor results in: And finally, multiplying the last terms in each factor: Algebraically adding all these terms in descending powers of x results in: That is exactly the expression you started with. So your answer is correct. Excellent job. Keep up the good work!
 Geometric_formulas/69579: Please help to identify the hypothesis and conclusion of the following statement: If you get a's on all your geometry tests, then everyone will want to sit near you in class.1 solutions Answer 49570 by bucky(2189)   on 2007-02-06 15:55:35 (Show Source): You can put this solution on YOUR website!This is a classic "If ... then" relationship where the "if" identifies the hypothesis (something supposed to be true for the sake of argument) and the "then" identifies the conclusion (the conclusion is the outcome of the hypothesis). Therefore, "if you get all a's on your geometry tests" is a hypothesis (something we suppose to be true) and "then everyone will want to sit near you in class" is the conclusion (outcome that will happen if the hypothesis is true). Notice that the hypothesis may never happen, but if it does, the conclusion could be wrong. For example, you may get all a's but maybe people don't like you ... or maybe they don't want the temptation of copying from you, so they may not want to sit by you anyway.
 Surface-area/69586: What is the area of a rectangle that has a perimeter of 47 mm and a length of 18.5 mm?1 solutions Answer 49568 by bucky(2189)   on 2007-02-06 15:37:07 (Show Source): You can put this solution on YOUR website!To find the perimeter (P) of a rectangle you need to add 2 times the length (L) to 2 times the width (W) of the rectangle. In equation form this is written: You are given that P is 47mm and L is 18.5mm. Substitute these values into the equation to get: Multiply out the 2*18.5 and the equation becomes: Subtract 37 from both sides to remove it from the left side of the equation. The result is: Finally, divide both sides by 2 and you find that: W = 5 mm. Then recognize that the area of a rectangle is found by multiplying its length times its width. You were given that the length was 18.5 mm and you found that the width was 5 mm. Multiplying these two together results in an area (A) equal to: which multiplies out to: and the units of the answer are square mm
 Equations/69590: 2(x+5)-x-4=71 solutions Answer 49567 by bucky(2189)   on 2007-02-06 15:21:09 (Show Source): You can put this solution on YOUR website!2(x+5)-x-4=7 First do the distributive multiplication by multiplying the two terms in the parentheses by the leading 2. This results in: 2x + 10 - x - 4 = 7 Next combine the 2x with the -x to get just x. When you do the equation becomes: x + 10 - 4 = 7 Then combine the +10 and the -4 to get +6 which makes the equation: x + 6 = 7 Then subtract 6 from both sides so that it no longer appears on the left side: x = +1 That is the answer. Check it by substituting +1 for x in the original problem to get: 2(1+5) - 1 - 4 = 7 Add the numbers in parentheses to get: 2(6) - 1 - 4 = 7 12 - 1 - 4 = 7 11 - 4 = 7 7 = 7 Since this is a true statement, the answer x = 1 is correct.
 Geometric_formulas/69588: Give a number that is a counter example to conjecture below. Explain your answer choice. All numbers that end in the digit 3 are prime.1 solutions Answer 49566 by bucky(2189)   on 2007-02-06 15:12:22 (Show Source): You can put this solution on YOUR website!The start of the series of numbers that end in 3 are: 3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, .... Numbers in this series that are not prime are 33 because it factors into 3 times 11; 63 because its factors include 21*3 and 9*7; and 93 because it can be factored into 31*3. Because these are just 3 of the counter examples, the premise that all digits that end in the number 3 are prime cannot be true.
 Linear-equations/69530: Hi, I need to find 3 solution for 6x-3/2y-12=12 Thanks1 solutions Answer 49532 by bucky(2189)   on 2007-02-06 02:32:12 (Show Source): You can put this solution on YOUR website!Given: Let's begin by simplifying the equation. Let's multiply everything by 2 to get rid of the denominator in the y-term. Now add 24 to both sides to move the -24 to the right side of the equal sign. Now all you have to do is to assume a value for x and find a corresponding value for y. For example, let's assume x = 1. The equation becomes: and this becomes: Subtract 12 from both sides to get: and by dividing both sides by -3 you find that y = -12 In the Cartesian point form (x,y) this means that (1, -12) is an answer. You can find other answers using the same process as above. For example, if you assume x = 2 and substitute that into the equation, you should find that y = -8 so that (2, -8) is another solution. And if you assume x = 4, you should find that y must = 0 so that (4, 0) is another solution. Another way to look at this is graphically. Return to the equation: First set x = 0 and the result will be y = 48/-3 = -16. Plot the point (0, -16) on the graph. Then let y = 0 and the result will be x = 48/12 = 4. Plot the point (4, 0) on the graph. Draw a line through those two points and extend the line in both directions. Any Cartesian point on that line will give you a set of x and y values that satisfies the equation. There are any number (an infinite amount) of values that will answer the problem. Hope this helps you understand the problem a little better.
 Inequalities/69532: Hello, What is the standard form for: y=3x-1? Thank you1 solutions Answer 49531 by bucky(2189)   on 2007-02-06 01:56:55 (Show Source): You can put this solution on YOUR website!The standard form of an equation with two variables (in this case the two variables are x and y) is: in which A, B, and C are constants with A and B not both zero. So what has to be done for this problem is to rearrange y = 3x -1 into the above form. Subtract 3x from both sides. If you do, you get: This simplifies to: If you compare this to the standard form, you will see that it is the same form in which A = -3, B = 1, and C = -1. Hope that helps you to see how to arrange linear equations with two variables into standard form.
 Miscellaneous_Word_Problems/69536: One share of A stock is worth 3 1/2 shares of B stock. If the total value of the stock is \$9000, how much was invested in each company?1 solutions Answer 49530 by bucky(2189)   on 2007-02-06 01:39:28 (Show Source): You can put this solution on YOUR website!One share of A stock is worth 3 1/2 shares of B stock. If the total value of the stock is \$9000, how much was invested in each company? This is a poorly designed question because it has no firm answer. For example, suppose a share of stock in company B is worth \$200. That means that a share of company A stock is worth 3 1/2 times as much or \$700. Now suppose that you own 2 shares of company A stock. That means you have \$1400 invested in company A. The remaining \$7600 (\$9000 minus \$1400) must be invested in company B and at \$200 per share the number of shares you have in company B is 7600/200 = 38 shares. Using the same value for the stocks, let's now suppose that you own 4 shares of company A stock at \$700 per share for a total of \$2800 invested in company A. The remaining \$6200 (\$9000 minus \$2800) must be invested in company B and at \$200 per share the number of shares you have in company B is 6200/200 = 31 shares. And a third example. At \$700 per share if you owned 8 shares of company A you would have a total of \$5600 invested in company A. That means the remaining \$3400 is invested in company B ... so you own 17 shares of company B. In these three examples a share of stock in company A is worth 3 1/2 times a share of stock in company B. And also in each example the total amount invested is \$9000. Those are all the limits set forth in the problem, so all three examples meet the requirements of the problem. Yet the amounts invested in each company differs for the in each example. There are many more examples you could come up with. If there were a limit on the number of shares that could be owned for each company or if the problem said that the total amount invested in company A was 3 1/2 times the total amount invested in company B that might make the problem practical. It is not a practical problem the way it is written. I think that what was intended in this problem was to say that the amount invested in company A (call it A) was 3 1/2 (use 3.5 in place of 3 1/2) times the amount invested in company B (call it B) and the total of the two investment amounts was \$9000. In such a case you can write: A = 3.5B and A + B = \$9000 Substitute 3.5B for A in the second equation 3.5B + B = \$9000 4.5B = \$9000 B = \$9000/4.5 = \$2000 And since A = 3.5B then A = 3.5*\$2000 = \$7000 But, again, that's not what the problem says as it is currently written.
 Surface-area/69359: how do you calculate the surface area of a cone with height of 18 ft and r of 10 ft1 solutions Answer 49412 by bucky(2189)   on 2007-02-05 09:28:53 (Show Source): You can put this solution on YOUR website!I presume that you are thinking of a right-circular cone ... that is, a cone whose height (h) is measured along a line perpendicular to the base that goes through the center of the circular base and through the vertex of the cone. The formula for this surface area is: where the variables in the equation are defined as: S is the surface area; r is the radius of the circle that forms the base; and s is the slant length of a line that goes from the perimeter of the circle forming the base to the vertex. By the way of explanation the length of s is the hypotenuse of a right triangle whose legs are r and h where h is the length of the perpendicular between the center of the circular base and the vertex. For your problem, you are given that the radius of the circular base is 10 feet and the height of the cone is 18 feet. From this calculate s using the Pythagorean theorem: s = s = s = s = 20.5913 ft Next substitute the known values of the variables into the surface area formula: When you do the multiplication you find that the surface area is 646.895 square feet.
 Proofs/69325: What would be the couterexample to the statement below? If the sum of two integers is even, then both of the integers are even.1 solutions Answer 49395 by bucky(2189)   on 2007-02-05 01:25:02 (Show Source): You can put this solution on YOUR website!If the sum of two integers is 8 (even), then the integers could be 5 and 3, and the two integers are both odd. You could do the same with the even sum of 12 being the sum of the odd integers 7 + 5 or 9 + 3.
 Linear-equations/69368: Is (1,-2) a solution of 3x-4y=12? Thank you1 solutions Answer 49394 by bucky(2189)   on 2007-02-05 00:52:13 (Show Source): You can put this solution on YOUR website!All you have to do is to recognize that (1,-2) is a way of saying that x = 1 and y = -2. You are given the equation 3x - 4y = 12 (1,-2) is a solution to this equation if you substitute 1 for x and -2 for y in the equation and the result is 12 = 12. In other words: 3x - 4y = 12 3(1) - 4(-2) = 12 3 + 8 = 12 11 = 12 Notice that the left side of the equation is not equal to the right side. Therefore, (1, -2) is not a solution to the equation.
 expressions/69375: f(x) = 2x-1 find f(-9)1 solutions Answer 49393 by bucky(2189)   on 2007-02-05 00:39:02 (Show Source): You can put this solution on YOUR website!You are given the expression: f(x) = 2x - 1 The notation f(-9) simply tells you that you substitute -9 for x in the expression f(x) = 2x-1 In this case the substitution of -9 for x leads to: f(-9) = 2(-9) - 1 = -18 - 1 = -19 so f(-9) = -19
 Exponents/69022: How do you simplify a cube of x to the fifth power1 solutions Answer 49200 by bucky(2189)   on 2007-02-03 00:34:54 (Show Source): You can put this solution on YOUR website!The cube of x is written Raising this to the 5 power is written as Notice that represents multiplied by itself 5 times ... or: **** and we multiply common terms with exponents by adding the exponents to get: **** = which simplifies to The shorthand way of doing this is to use the rule that when a term with an exponent is raised to another exponent, you multiply the two exponents. In this case: is found by multiplying the 3 by the 5 to get Hope this helps you to understand exponents a little more.
 Numbers_Word_Problems/69102: Using the numbers 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, find four consecutive numbers whose sum is one half the sum of the other eight numbers.1 solutions Answer 49197 by bucky(2189)   on 2007-02-02 23:54:35 (Show Source): You can put this solution on YOUR website!If you assume that the first integer is N, then the next three integers are N+1, N+2, and N+3. You need to know the sum of these four integers in a row. The sum is N+(N+1)+ (N+2)+(N+3). Since all the parentheses are preceded by a + sign, the parentheses can be removed without changing any of the terms in the parentheses. The result is: N+N+1+N+2+N+3 and by adding like terms this sum is simplified to 4N+6 Next you need to find the sum of the remaining 8 numbers. You can get this by adding all 12 numbers and then subtracting the sum of the four consecutive numbers we found above. Probably the quickest way to find the sum of all 12 numbers is to just add 21 through 32 on a calculator. You can also find it by multiplying 12 times (21+32) and dividing that answer by 2. [This method involves the rule that to add consecutive numbers in a series, add the first number in the series to the last number in the series, multiply that sum by the number of numbers in the series, and divide that result by 2.] In either case you should find that the sum of the 12 consecutive numbers in the series whose first number is 21 and whose last number is 32, is a total of 318. If you subtract from 318 the sum of the four consecutive numbers, the remainder will be the sum of the remaining eight numbers. Since the sum of the 4 consecutive numbers is 4N+6, the subtraction is: 318-(4N+6) = 318-4N-6 = 312-4N This is the sum of the other eight numbers The problem tells you that the sum of the four consecutive numbers equals one-half the sum of the remaining eight numbers. In equation form this is: 4N+6 = (1/2)
 Numbers_Word_Problems/69104: Assume you have the typical heartbeat rate. If you take your pulse for 10 seconds, how many beats would you feel?1 solutions Answer 49192 by bucky(2189)   on 2007-02-02 23:07:54 (Show Source): You can put this solution on YOUR website!Were you told what the typical heartbeat rate is? Some sites list it as 72 beats per minute for adult males and 76 to 80 beats per minute for adult females. Another site lists the spectrum of heart beats as 60 to 110 with an average of around 80 for ages 12 through 16 and as 50 to 90 beats per minute with an average of 70 for persons greater than 16 years of age. These rates are for persons that are calm and at rest without stress. Obviously if the person is undergoing aerobic exercises, the heartbeat rate will increase significantly. Notice that the rates are expressed as the number of beats per minute. If you count the beats for 10 seconds, you have only counted for one-sixth of a minute. So you must multiply the number of beats in that 10 seconds by 6 times to find the number of beats per minute. And going in the other direction, if you assume that the typical heartbeat rate is 72 beats per minute, then you can determine the number of beats in 10 seconds by dividing the 72 beats per minute by 6 because there are 6 each 10-second intervals in a minute. So the answer is 72/6 = 12 beats in 10 seconds. If you were given a different heartbeat rate, divide it by 6 to find how many beats you would count in 10 seconds. For practice you should assume different average heartbeat rates and determine how many beats would occur in 10 seconds. For example, if the average heartbeat rate for a teenager is 80 beats per minute, how many beats would be counted in 10 seconds? The answer is about 13 beats in 10 seconds, but how do you determine that? Or suppose that a person has a heartbeat rate of 68 beats per minute. How many beats would you count in 15 seconds? Since there are 4 periods of 15 seconds each in a minute, each period would get one-fourth of the 68 beats that occur in an entire minute. So the answer can be obtained by dividing 68 by 4. Nurses often count the number of beats in 15 seconds and multiply that number by 4 to find the heartbeat count in beats per minute for a patient. Hope this helps you to understand the problem a little better.
 Complex_Numbers/69072: Solve (1 + i) / (1 - i) Note how parentheses have been added to make the numerator (1+i) and the denominator (1-i) which is what I think you intended.1 solutions Answer 49160 by bucky(2189)   on 2007-02-02 16:53:49 (Show Source): You can put this solution on YOUR website!Any time that you have a complex number in the denominator look to change it to a real number by multiplying it by its complex conjugate ... which is to say multiply it by itself except with a change in the sign between the real and imaginary portions of the complex number. Easier to understand if we work this problem. In the denominator there is the term 1-i. Multiply it by 1+i. (This is the complex conjugate of 1-i ... same terms only change of the sign between the terms.) Recall from algebra that: (a-b)(a+b) = a^2 - b^2. Thats exactly what we have except a = 1 and b=i. So the product of (1-i)(1+i) = 1^2 -(i^2) = 1 - i^2. But also recall that by definition i^2 = -1. So our multiplication which resulted in 1 - i^2 becomes 1 - (-1) when we substitute -1 for i^2. And 1 - (-1) becomes 1 +1 = 2. Therefore, after multiplying the original denominator of 1-i by its conjugate of 1+i the denominator is converted to 2. However, we can't multiply the denominator by 1+i without also multiplying the numerator by 1+i. [In effect that is equivalent to multiplying the original fraction by (1+i)/(1+i) which is to say, multiplying the original fraction by 1.] Since the original fraction had 1+i and the numerator, we multiply that by 1+i (which is to say that we square the numerator in this case). The result is: (1+i)*(1+i) = 1^2 + 1i + 1i + i^2. This simplifies to 1+2i+i^2, but recall that i^2 equals -1 by definition. Therefore we substitute -1 for i^2 to get 1+2i-1. The real parts cancel and we are left with just 2i in the numerator. We now divide this by the denominator of 2 to get 2i/2 = i. The answer is just i. Hope this helps you to think about complex number division