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Equations/628135: I have some equations in the following format (finding the nth term of an algebraic expression) ( n ) a^(n-(r-1)) * b^(r-1) r-1 I don't understand what the (n & r-1) terms in the parenthesis at the beginning of the expression are. From the solution, it suggests that the first term ends up resolving to a division of factorials such as: (6! / ((6-3)! * 3!)) where n=6 and r=4... I just dont understand how they got from the first form to the factorial form. 1 solutions Answer 395467 by solver91311(17077)   on 2012-07-13 16:15:20 (Show Source): You can put this solution on YOUR website! I think what you are looking at is (read: "n choose r - 1") which is the number of combinations of things taken at a time. Such as, how many ways can I choose from 10 different books on a shelf if I take them 3 at a time and I don't care what order the three are in when I get them in my hand? In general, is calculated by . Hence, your is exactly correct given and . Here's the logic: Let's say you have 6 things and want to choose 3 of them. There are 6 ways to choose the first one, then since you didn't replace the first one you chose, there are 5 ways to choose the second one for each one of the 6 ways to choose the first one. Then there are 4 ways to choose the third one for each of the 30 ways to pick the first two, which works out to which comes from . But that number is too large by a factor of the number of ways to arrange the three things in your hand, namely , and that is where the other denominator factor comes from. You might want to compare this to the number of permutations of things taken at a time. With permutations, order matters. Such as you have 20 people in your club and you want to know how many different ways you can select a President, Secretary, and Treasurer. Here order matters because Suzy being the president is a different outcome than Suzy being the Secretary, for example. Permutations are calculated . See the difference? By the way, this is also the th coefficient (counting from zero) of the binomial expansion of and the th element of the th row of Pascal's Triangle. John My calculator said it, I believe it, that settles it

Travel_Word_Problems/628193: two airplanes leave the airport at the same time, one heading west the other east. their average speeds differ by 10 miles per hour. after 1.5 hours, they are 520 miles apart. what is the approximate average speed of each plane over the 105 hours? 1 solutions Answer 395449 by solver91311(17077)   on 2012-07-13 14:10:23 (Show Source): You can put this solution on YOUR website! Divide the total distance by the given amount of time to get the SUM of the rates. Let be the rate of one plane and then must be the rate of the other plane. Add these two quantities: and set this equal to the sum rate you calculated at the beginning. Solve for and then calculate John My calculator said it, I believe it, that settles it

�t�e�s�t/628195: A RANDOM SAMPLE OF 120 STUDENTS HAS A TEST SCORE AVERAGE WITH A STANDARD DEVIATION OF 9.2. FIND THE MARGIN OF ERROR IF C=0.98. 1 solutions Answer 395448 by solver91311(17077)   on 2012-07-13 14:03:50 (Show Source): You can put this solution on YOUR website! First write 100 times: "I will not type in all caps because it is shouting and therefore both rude and annoying." Then I'll help you. John My calculator said it, I believe it, that settles it

test/628198: IF THE PROBABLITY OF A NEWBORN CHILD BEING FEMALE IS 0.5 FIND THE PROBABLITY THAT IN 100 BIRTHS 55 OR MORE WILL BE FEMALE. USE THE NORMAL DISTRIBUTION TO APPROXIMATE THE BINOMIAL DISTRIBUTION 1 solutions Answer 395447 by solver91311(17077)   on 2012-07-13 14:03:33 (Show Source): You can put this solution on YOUR website! First write 100 times: "I will not type in all caps because it is shouting and therefore both rude and annoying." Then I'll help you. John My calculator said it, I believe it, that settles it

test/628197: ASSUME THAT HEIGHTS OF MEN ARE NORMALLY DISTRIBUTED WITH A MEAN OF 68.4 INCHES AND A STANDARD DEVIATION OF 2.8 INCHES. IF 64 MEN WERE RANDOMLY SELECTED FIND THE PROBABILITY THST THEY HAVE A MEAN HEIGHT GREATER THAN 69.4 INCHES 1 solutions Answer 395445 by solver91311(17077)   on 2012-07-13 14:03:18 (Show Source): You can put this solution on YOUR website! First write 100 times: "I will not type in all caps because it is shouting and therefore both rude and annoying." Then I'll help you. John My calculator said it, I believe it, that settles it

Linear-systems/628183: how do i solve 3x + 7y = 16 by using the addition method 4x - 3y = 9 1 solutions Answer 395437 by solver91311(17077)   on 2012-07-13 13:32:38 (Show Source): You can put this solution on YOUR website! Multiply the first equation by 3 and the second one by 7. Add the two equations term-by-term to eliminate the -variable. Solve the resulting single variable equation for . Substitute the derived value of back into either of the original equations then solve for Correct your vocabulary. The Elimination method is more accurate and descriptive than the Addition method. John My calculator said it, I believe it, that settles it

Geometry_Word_Problems/628171: What is the midpoint of a-b given that a is(4,-3) and b is(-2,-11) 1 solutions Answer 395435 by solver91311(17077)   on 2012-07-13 13:27:18 (Show Source): You can put this solution on YOUR website! Use the midpoint formulas to calculate each of the coordinates of the midpoint: and where and are the coordinates of the given points. John My calculator said it, I believe it, that settles it

Probability-and-statistics/628182: A set of 50 data values has a mean of 15 and a variance of 36. Find the standard score of a data value = 18. 1 solutions Answer 395431 by solver91311(17077)   on 2012-07-13 13:21:03 (Show Source): You can put this solution on YOUR website! Subtract the mean from the data point and divide the difference by the positive square root of the variance. John My calculator said it, I believe it, that settles it

Geometry_Word_Problems/628180: If a rectangle with length L and width W has a perimeter of 4 cm, what are the dimensions of the rectangle with the largest area and what is that area? 1 solutions Answer 395428 by solver91311(17077)   on 2012-07-13 13:13:14 (Show Source): You can put this solution on YOUR website! The perimeter of a rectangle is given by: Which can be rearranged to: The area, given by length times width, can then be represented as a function of the width for a given perimeter thus: This function is a quadratic that can be put into standard form thus: The graph of such a quadratic is a parabola that is concave down (the lead coefficient is less than zero), meaning that the vertex represents a maximum. The coordinate of the vertex of this parabola is given by: Therefore the width that gives the greatest area rectangle for any given perimeter is the perimeter divided by 4. The two widths are therefore half of the perimeter, hence the two lengths must also be half of the perimeter, and therefore the shape is a square. Divide your given perimeter by 4 and then square the result. John My calculator said it, I believe it, that settles it

Linear-equations/628181: how do i use the given conditions to write an equation for the line in point slope form slope=4, passing through(-3,7) 1 solutions Answer 395427 by solver91311(17077)   on 2012-07-13 13:10:59 (Show Source): You can put this solution on YOUR website! Use the point-slope form: where are the coordinates of the given point and is the given slope. All you have to do is plug in the numbers. John My calculator said it, I believe it, that settles it

Functions/628172: How can I find the range of x/(x+2)? 1 solutions Answer 395426 by solver91311(17077)   on 2012-07-13 13:07:54 (Show Source): You can put this solution on YOUR website! Look at the behavior of the function as and as . Then look at the behavior as from both the right and the left. Consider the sign of the function value on the following intervals: , , and Remember, the horizontal asymptote of a rational function where the numerator an denominator are of equal degree is where is the lead coefficient of the numerator and is the lead coefficient of the denominator. John My calculator said it, I believe it, that settles it

Human-and-algebraic-language/628072: Must be in Algebraic Expression with Correct Units for the following problem: _____________________________________________________________________________ The length of the SIDE of a Rectangular desk is; x-4y cm. The length of the front is; x+4y cm. What is the area of the TOP of the rectangular desk? _____________________________________________________________________________ All help greatly appreciated! 1 solutions Answer 395373 by solver91311(17077)   on 2012-07-12 22:59:00 (Show Source): You can put this solution on YOUR website! Multiply the two binomial expressions. Note that they are a conjugate pair, so the product is the difference of two squares. John My calculator said it, I believe it, that settles it

Conjunction/628087: Construct a truth table for (~p ᴧ q) → p, be sure to include all your work 1 solutions Answer 395372 by solver91311(17077)   on 2012-07-12 22:56:58 (Show Source): You can put this solution on YOUR website! Can't. Don't know what ᴧ means. John My calculator said it, I believe it, that settles it

test/628088: 1 solutions Answer 395371 by solver91311(17077)   on 2012-07-12 22:55:51 (Show Source): You can put this solution on YOUR website!

test/628089: 1 solutions Answer 395370 by solver91311(17077)   on 2012-07-12 22:55:29 (Show Source): You can put this solution on YOUR website!

test/628091: 1 solutions Answer 395369 by solver91311(17077)   on 2012-07-12 22:55:13 (Show Source): You can put this solution on YOUR website!

Exponents/628009: The exact solution(s) of 5^(5x − 3) = 8^(−x − 1) are the following. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) x = (?,?) 1 solutions Answer 395368 by solver91311(17077)   on 2012-07-12 22:54:45 (Show Source): You can put this solution on YOUR website! I just finished doing one just like this for you. Follow the same procedure with the different numbers. John My calculator said it, I believe it, that settles it

Exponents/628010: Use algebraic procedures to find the exact solution(s) of the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 4^(4x − 3) = 5^(−x − 1) x=(?,?) 1 solutions Answer 395366 by solver91311(17077)   on 2012-07-12 22:40:13 (Show Source): You can put this solution on YOUR website! Take the log of both sides (any base, doesn't matter) Use on both sides Multiply both sides by just for the sake of neatness, let Then it is just plain old algebra from here: Since you want the exact answer, and you don't care about actually being able to approximate the log values, you can get a much neater answer using base 4 logs: I'll let you figure out why these two answers are the same. Hint: What is the value of ? John My calculator said it, I believe it, that settles it

Travel_Word_Problems/628082: john traveled 40mph for half of his jorney and the other half at 60mph what was his average speed? show me the steps please 1 solutions Answer 395360 by solver91311(17077)   on 2012-07-12 22:14:50 (Show Source): You can put this solution on YOUR website! It depends on what you mean by "half of his jorney[sic]" If you mean half the distance, then you get one answer, but if you mean half the time, you get another answer. So which is it? John My calculator said it, I believe it, that settles it

Unit_Conversion_Word_Problems/628084: I have 3400 pounds and get paid $8.75 per hundred pound how much would i make 1 solutions Answer 395359 by solver91311(17077)   on 2012-07-12 22:07:29 (Show Source): You can put this solution on YOUR website! Divide the number of pounds you have by 100, then multiply the result by the price per hundredweight. John My calculator said it, I believe it, that settles it

Travel_Word_Problems/628085: Two trains leave chicago, one headed due east at 60 mph and one headed due west at 50 mph. How long before they will be 990 miles apart? 1 solutions Answer 395358 by solver91311(17077)   on 2012-07-12 22:06:05 (Show Source): You can put this solution on YOUR website! Since they are traveling in opposite directions, add their speeds. Then divide distance by total speed to get time. John My calculator said it, I believe it, that settles it

Trigonometry-basics/628076: How do you determine the range of this function: y = 2 + 2sec(2x) 1 solutions Answer 395355 by solver91311(17077)   on 2012-07-12 21:59:51 (Show Source): You can put this solution on YOUR website! Just looking at the secant function, and it has no upper bound. Since the absolute value of secant can be no less than 1, Range: John My calculator said it, I believe it, that settles it

Geometry_Word_Problems/628074: A rectangular piece of metal is 30 in longer than it is wide. Squares with sides 6 in. long are cut from four corners and the flaps are folded upward to form on open box. If the volume of the box is 594 cubic inches, what were the original dimensions of the piece of metal? 1 solutions Answer 395351 by solver91311(17077)   on 2012-07-12 21:33:32 (Show Source): You can put this solution on YOUR website! If the sides of the cutout squares are 6", then the depth of the box when the sides are folded up must be 6 inches. Divide the volume by 6 to get the area of the bottom of the box, 99 square inches. The original piece of metal had a width of and therefore had a length of Since you took 6 X 6 inches out of each corner, subtract 12 inches from each of the width and the length to get expressions for the width and length of the bottom of the box. Expand the binomials, put the quadratic into standard form, and then solve for . Then calculate John My calculator said it, I believe it, that settles it

Mixture_Word_Problems/628075: Essay: Must SHOW ALL WORK! You want to create a rectangular garden, its length is 3x-3y ft. and x-5y ft. in width. Be sure to multiply this problem out class, and express it in simplest Correct mathematical form including units. _______________________________________________________________________________ Thank you in advance. 1 solutions Answer 395349 by solver91311(17077)   on 2012-07-12 21:20:27 (Show Source): You can put this solution on YOUR website! You haven't asked a question. What do you need to know? What is it that you want to calculate? The area? The perimeter? John My calculator said it, I believe it, that settles it

Equations/628054: How do you Use an algebraic approach to solve the problem. A board 45 feet long is cut into two pieces such that the length of one piece is two-thirds of the length of the other piece. Find the length of the shorter piece of board. so far what i have is 2/3x=45 is this right? if so how do you solve? 1 solutions Answer 395337 by solver91311(17077)   on 2012-07-12 20:42:16 (Show Source): You can put this solution on YOUR website! Let's see if your approach makes sense. You said . If a part of the quantity is equal to 45, then must be bigger than 45. If I asked you for two-thirds of the money in your wallet and you handed me $45, you must have started with more than the 45 bucks, right? That means, according to your theory, that one of the pieces of board is longer than the piece it was cut from. Absurd in the extreme. Read the problem carefully. I'm going to assign to represent the measure of the longer piece, just because it is easier to explain the problem that way. Generally you will want to assign your variable to represent the quantity asked for by the problem, but that is just a little inconvenient this time. Let represent the measure of the longer piece. Then we are given that is the measure of the shorter piece. Together (except for a negligible saw kerf) the two boards should add up to the original 45 feet. Solve for and then calculate Step 1: Multiply both sides by 3: Step 2: Collect like terms: Step 3: Multiply by the reciprocal of the coefficient on the variable: Which is the measure of the larger piece. You can either calculate or to find the measure of the shorter piece. John My calculator said it, I believe it, that settles it

�t�e�s�t/628013: Please help me with this question: Suppose that y=5x+2 and we want y to be within 0.07 unit from 6. for what values of x will this be true? Thank you Samantha 1 solutions Answer 395325 by solver91311(17077)   on 2012-07-12 19:59:22 (Show Source): You can put this solution on YOUR website! Solve the absolute value inequality for the solution set interval. John My calculator said it, I believe it, that settles it

Functions/628028: If 10 is in the domain of f−1, find f [f−1(10)]. f[f−1(10)] = 1 solutions Answer 395321 by solver91311(17077)   on 2012-07-12 19:51:46 (Show Source): You can put this solution on YOUR website! If is in the domain of then John My calculator said it, I believe it, that settles it

Quadratic_Equations/628033: 1. Suppose you are an event coordinator for a large performance theater. One of the hottest new Broadway musicals has started to tour and your city is the first stop on the tour. You need to supply information about projected ticket sales to the box office manager. The box office manager uses this information to anticipate staffing needs until the tickets sell out. You provide the manager with a quadratic equation that models the expected number of ticket sales for each day x. ( is the day tickets go on sale). - 0.4x2 + 13x + 14 a. Does the graph of this equation open up or down? How did you determine this? b. Describe what happens to the tickets sales as time passes. c. Use the quadratic equation to determine the last day that tickets will be sold. Note. Write your answer in terms of the number of days after ticket sales begin. d. Will tickets peak or be at a low during the middle of the sale? How do you know? e. After how many days will the peak or low occur? f. How many tickets will be sold on the day when the peak or low occurs? g. What is the point of the vertex? How does this number relate to your answers in parts e. and f? h. How many solutions are there to the equation? - 0.4x2 + 13x + 14. How do you know? i. What do the solutions represent? Is there a solution that does not make sense? If so, in what ways does the solution not make sense? 1 solutions Answer 395317 by solver91311(17077)   on 2012-07-12 19:45:06 (Show Source): You can put this solution on YOUR website! This same question (though with slightly different coefficients in the function) has been answered 40 or 50 times on this site. Click on Algebra.com and then type "event coordinator" in the little search box. Pick a solution from several pages of them and then adjust the numbers. John My calculator said it, I believe it, that settles it

test/628003: Find a polynomial f(x) with real coefficients having the given degree and zeros. Degree 5; zeros: 6;-i;-7+i f(x)=a? Do not write the polynomial in factored form. Multiply it out. 1 solutions Answer 395294 by solver91311(17077)   on 2012-07-12 17:44:17 (Show Source): You can put this solution on YOUR website! If is a zero of a polynomial, then is a factor of the polynomial. Complex zeros of polynomials with real coefficients always appear in conjugate pairs. You didn't specify that your function has real coefficients, but unless that is true it is impossible to solve the problem as posed. Hence, I make the assumption of real coefficients. is a real zero, so is a factor of the polynomial. The conjugate of a general complex number is formed by changing the sign on the second term thus: is a complex zero. To form the conjugate of , rewrite it as , then it should be clear that the conjugate is which can be written as simply . Likewise your other zero, , is a complex number, hence the fifth and last zero of your 5th degree polynomial is Now just form the factors: Now the only work left for you to do is to multiply out the five binomials. Note that the product of two conjugate binomials is the difference of two squares, but because , the product of two complex conjugates becomes the sum of two squares. John My calculator said it, I believe it, that settles it

Triangles/627928: Which three lengths could be the sides of a triangle: 8,15,22 22,5,10 11,6,17 6,22,10 1 solutions Answer 395219 by solver91311(17077)   on 2012-07-12 12:04:05 (Show Source): You can put this solution on YOUR website! The sum of the measures of any two sides of a triangle must be strictly greater than the measure of the third side. Go through your list. For each of the sets of three numbers, find the two smallest. Add them. If the sum is less than or equal to the largest number, then the set of numbers does NOT represent the measures of the sides of a triangle. John My calculator said it, I believe it, that settles it

Geometry_Word_Problems/627929: The perimeter of a rectangular lot is 56 meters. The length exceeds the width by 12 meters. Find the length of the lot 1 solutions Answer 395217 by solver91311(17077)   on 2012-07-12 11:56:56 (Show Source): You can put this solution on YOUR website! If a rectangle with perimeter has a length that is units longer (or shorter if ) than times the width, , Then knowing that the Perimeter is given by: , make the substitution: Then solve for For your problem: , , and . Do the arithmetic. John My calculator said it, I believe it, that settles it