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# Recent problems solved by 'Theo'

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 logarithm/674335: graph this function g(x)=log2of(x+1)1 solutions Answer 419180 by Theo(3458)   on 2012-10-31 15:52:03 (Show Source): You can put this solution on YOUR website! i appears there's a vertical asymptote at x = -1. the following table helps show that. ```x y = log(2,x+1) -2 #NUM! -1 #NUM! -0.9999 -16.60964047 -0.9 -3.321928095 -0.8 -2.321928095 -0.7 -1.736965594 -0.6 -1.321928095 -0.5 -1 0 0 1 1 2 1.584962501 3 2 4 2.321928095 5 2.584962501 6 2.807354922 7 3 8 3.169925001 9 3.321928095 ``` you cannot take a log of a negative number of or 0. that is why the first 2 values in the table are invalid. that is why the graph stops as just to the right of x -1 log (x+1) will be log(0) when x is equal to -1.
 Travel_Word_Problems/674325: Could you help me answer this question? A plane flies on a heading/bearing of 75 degrees at 460 mph. To the nearest hundredth how far is the plane from the airport at the end of 3.2 hours? How far east of the airport? How far north of the airport? I calculated that it would travel 1472 miles in 3.2 hours. 1 solutions Answer 419176 by Theo(3458)   on 2012-10-31 15:37:25 (Show Source): You can put this solution on YOUR website!3.2 * 460 = 1472 which means the plan has traveled 1472 miles in 3.2 hours at 460 mph. that part is correct. the rest is trigonometry. make a graph and then make a line that makes an angle of 75 degrees with the x-axis. the line starts at the origin of the graph and extends upward to the right at the 75 degree angle from the x-axis. you now can make a right triangle. the hypotenuse is 1472 miles long. the horizontal leg of this triangle represents the number of miles he travels in an easterly direction and the vertical leg of this triangle represents the number of miles he travels in a northerly direction. the horizontal distance he travels we'll call x the vertical distance he travels we'll call y the hypotenuse we'll call z. since this is a right triangle, then we get the following trigonometric relationships. sin(75) = y/1472 cos(75) = x/1472 we use these equations to solve for x and y y = 1472 * sin(75) x = 1472 * cos(75) use your calculator to solve for x and y you should get: y = 1421.842816 x = 380.9816344 x is the number of miles he traveled in an easterly direction. y is the number of miles he traveled in a northerly direction. z is the number of miles he traveled in a combined easterly and northerly direction at a 75 degree angle from the east. the following diagram shows you what i mean. in the diagram, the numbers are rounded. z = 1472 x = 381 y = 1422 unfortunately, 1422 looks a lot like 1472 but if you look closely there is a difference. just bad handwriting on my part - sorry for that.
 Age_Word_Problems/674243: Bob spent 1/6 of his life as a child, 1/12 as an adolescent, and 1/7 as a bachelor, five years after he was married, he had a son who died 4 years before his father at half his father's final age. How long did bob live?1 solutions Answer 419158 by Theo(3458)   on 2012-10-31 14:34:20 (Show Source): You can put this solution on YOUR website!he spent 1/6 of his life as a child, 1/12 as an adolescent, and 1/7 as a bachelor. 5 years after he's married, he has a son. son dies 4 years before his father at half his father's final age. how long did he live? if we add up 1/6 and 1/12 and 1/7, we get 33/84. this should mean that he dies at 84 and has 51 years left to live. if that holds up, then we found the answer. let's see if it does. assuming we're right, then he gets married at 33 years of age. 5 years later he has a son when he is 38 years old. based on our assumption, he dies at 84. his son has died 4 years earlier at half his age when he dies which means his son has died at age 42. here's the timeline. ```his age his son's age description of event 38 0 his son is born. 80 42 his son dies. 84 he dies. ``` when his son dies at 42, he is 80 (80 - 38 = 42) 4 years later he dies at 84. numbers appear to check out so the solution is good. it appears the only information we needed was to be able to add up 1/6 + 1/12 and 1/7 to find out how much of his life was consumed when he got married. the remainder from the denominator is what he had left. the denominator is how long he lived.
 Linear_Equations_And_Systems_Word_Problems/674300: A new candle is 8 inches tall and burns at a rate of 2 inches per hour. a. Write an equation that models the height h after t hours. b. Sketch the graph of the equation. 1 solutions Answer 419142 by Theo(3458)   on 2012-10-31 13:15:26 (Show Source): You can put this solution on YOUR website!y = -2x + 8 y is the height of the candle in inches. x is the amount of time it burns in hours. in 4 hours the candle should burn down to nothing. domain of the function is x = 0 to 4. range of the function is y = f(x) = 0 to 8 graph of the function looks like this:
 Triangles/673974: Hi I need help on how to find the perimeter of a triangle when two angles are congruent and it includes exponents, so for triangle DEF angles D and F are congruent, DE= 10x^2-100 EF= 5x^2+40x and DF=15x. Thank you! 1 solutions Answer 418964 by Theo(3458)   on 2012-10-30 18:45:15 (Show Source): You can put this solution on YOUR website!triangle is DEF Angle D and F are congruent. this means that sides opposite those angles are also congruent. that would be sides DE and EF the length of side DE is 10x^2 - 100 the length of side EF is 5x^2 + 40x set these sides equal to each other and you get: 10x^2 - 100 = 5x^2 + 40x subtract 5x^2 and 40x from both sides of the equation and you get: 10x^2 - 100 - 5x^2 - 40x = 0 combine like terms and you get: 5x^2 - 40x - 100 = 0 factor out a 5 and the equation becomes: x^2 - 8x - 20 = 0 factor this to get: (x-10) * (x+2) = 0 solve for x to get x = 10 or x = -2. since x can't be negative, then x = 10 is the solution you are looking for. now that you know that x = 10, you can solve for the perimeter by replacing x with 10 in each equation and then adding up the result. your equations are: 10x^2 - 100 5x^2 + 40x 15x when x = 10, these equations become 10*10^2 - 100 = 1000 - 100 = 900 5*10^2 + 40*10 = 5*100 + 400 = 500 + 400 = 900 15*10 = 150 the sum of all of these is 1800 + 150 which gets you a perimeter of 1950.
 Polynomials-and-rational-expressions/673800: Solve the following inequality: > 01 solutions Answer 418957 by Theo(3458)   on 2012-10-30 18:19:36 (Show Source): You can put this solution on YOUR website!you can solve this by graphing the equation and you can then visually see the regions that the solution lies in. the graph looks like this. it confirms the algebraic solution provided by edwin. the values between -1 and 1 are not mathematically possible and therefore don't show up on the graph.
 Quadratic-relations-and-conic-sections/673675: Focus is 0,2 and the directrix is y= -4 How do I find the equation of this parabola is the directrix the same as the y intercept? given the vertex and the focus how do I find the equation of the parabola? I know I use the distance formula for these, can you show me where to put the given numbers in the distance formula and I can keep these as samples because I have a lot of these to do. I also have x=2 for a directrix. Where would I put that in the formula? Thank you!1 solutions Answer 418896 by Theo(3458)   on 2012-10-30 14:15:33 (Show Source): You can put this solution on YOUR website!focus is 0,2 and the directrix is y = -4 the distance between the focus and the directrix is equal to (0,2) to (0,-4) which is equal to 6 that makes 2p = 6 which makes p = 3 there are 3 possible equations for this parabola. they are: ax^2 + bx + c = 0 (standard form) y = a(x-h)^2 + k (vertex form) y = (x-h)^2/4p + k (conics form) another form of the equation in the conics form is: 4p(y-k) = (x-h)^2 we should be able to work either either one because they are equivalent. we should be able to solve for all of them. let's see if we can. since p = 3, then 4p in the conics form must be equal to 12. since a = 1/4p, this mean that a in the vertex form must be equal to 1/12. (h,k) is the vertex of the parabola. since you know the focus, then you can solve for the vertex because the vertex is halfway between the focus and the directrix. you're going from (0,2) to (0,-4) which gets you to a vertex of (0,-1) because that is halfway between (0-2) and (0,-4). with a vertex of (0,-1), we get h = 0 and k = -1. we may be able to piece together some of the equations now. we have: h = 0 k = -1 4p = 12 in the conics form a = 1/12 in the vertex form note that a in the vertex form is not the same as a in the standard form. we'll start with the conics form. the equation is: y = (x-h)^2/4p + k (conics form) substitution gets us: y = x^2/12 - 1 let's try the vertex form next. the equation is: y = a(x-h)^2 + k (vertex form) substitution gets us: y = 1/12 * x^2 - 1 which is equivalent to: y = x^2/12 - 1 this is the same as the conics form so we're doing good so far. let's try the standard form next. the equation is: ax^2 + bx + c = 0 (standard form) since we know that the equation is y = x^2/12 - 1, we just set y = 0 and we get: x^2/12 - 1 = 0 in standard form, this means that: a = 1/12, b = 0, and c = -1 it's not clear to me if the a in the standard form is the same as the a in the vertex. it does seem to work out in this example so maybe it is. a graph of this equation of y = x^1/12 - 1 is shown below: from the graph, you can see that the focus is at (0,2) and the vertex is at (0,-1) and the point on the line y = -4 that you would measure to is the point (0,-4) which is in line with the focus and the vertex and lies on the axis of symmetry which is x = 0. the reference that i used to answer this question is at: http://www.purplemath.com/modules/parabola.htm this is an excellent reference and well worth reading to understand this better. it's a little difficult to follow but just be careful of all the a's and b's and c's and p's and you should be able to figure it out. be careful though because they talk about the standard equations that have the parabola facing up or down, and the sideways equations that have the parabola facing left or right. ideally, you should be understand both, but for this problem you only need to understand the up and down equations. those are the ones i showed you above. i got them from this reference so they should track with what the reference is saying. the reference is 4 pages long. what you see when you click on the link is the first page only. good luck.
 test/673759: The volume of gas varies directly as its temperature and inversely as its pressure. At a temperature of 100 kelvin and a pressure of 15 kilograms per square meter, the gas occupies a volume of 20 cubic meters. Find the volume at a temperature of 150 kelvin and a pressure of 20 kilograms per square meter.1 solutions Answer 418852 by Theo(3458)   on 2012-10-30 12:36:50 (Show Source): You can put this solution on YOUR website!direct proportion is z = kx inverse proportion is z = k/y direct and inverse proportion together is z = kx/y z = volume of gas in cubic meters x = temperature in kelvin y = pressure in kilograms per square meter. when volume of gas is 20 cubic meters and temperature is 100 kelvin and pressure is 15 kilograms per square meter, equation of z = kx/y becomes 20 = 100k/15. solve for k to get k = 20*15/100 which results in k = 3. now that you know k, you can use that to solve the problem. when the temperature is 150 kelvin and the pressure is 20 kilograms per square and k = 3, the equation of z = kx/y becomes z = 3*150/20 which becomes z = 22.5 cubic meters. that's your answer.
 test/673750: Which of the following patterns is determined from the expression below? 1/6 + 1/9n Here are the possible answers 5/18,4/9,11/18, 7/9 or 1/6,5/18,7/8,1/2 or 5/18,7/18,1/2, 11/18 or 1/18,-1/18, -1/6, -5/8. How do you figure this out??? Please help1 solutions Answer 418842 by Theo(3458)   on 2012-10-30 12:13:56 (Show Source): You can put this solution on YOUR website!the answer is: 5/18, 7/18, 1/2, 11/18 your equation looks like it is (1/6 + (1/9)*n n goes from 1 to ........ the first term has n = 1 the second term has n = 2 etc. when n = 1, the equation is 1/6 + 1/9 which results in 5/18 when n = 2, the equation is 1/6 + 2/9 which results in 7/18 when n = 3, the equation is 1/6 + 3/9 which results in 9/18 which results in 1/2 when n = 4, the equation is 1/6 + 4/9 which results in 11/18 if you look at your selections, the 4th selections appears to be it. that reads: 5/18 followed by 7/18 followed by 1/2 followed by 11/18.
 Age_Word_Problems/673733: Suppose that Kristin ate two McDonald hamburgers and drank one meduim Coke for a total of 770 calories. Kristins friend Jack ate three hamburgers and drank 2 Cokes for a total of 1260 calories. How many calories are in a McDonalds hamburgers? How many calories are in a meduim coke? 1 solutions Answer 418834 by Theo(3458)   on 2012-10-30 12:00:27 (Show Source): You can put this solution on YOUR website!let x = munber of hamburgers let y = number of medium cokes. your equations become: 2x + y = 770 3x + 2y = 1260 solve for y in the first equation to get y = 770 - 2x substitute for y in the second equation to get: 3x + 2*(770 - 2x) = 1260 simplify to get: 3x + 1540 - 4x = 1260 combine like terms and subtract 1540 from both sides of the equation go get: -4x = 1260 - 1540 which becomes: -x = -280 multiply both sides by -1 to get: x = 280 the number of calories in a mcdonald's hamburger is 280. replace x with 280 in either equation and solve for y. we'll use the equation of: 2x + y = 770 replace x with 280 and you get: 560 + y = 770 subtract 560 from both sides to get: y = 770 - 560 which becomes: y = 210 the number of calories in a medium code is 210. you get: x = 280 y = 210 x = number of calories in a hamburger. y = number of calories in a medium coke. 2 hamburgers and 1 medium coke is 2 * 280 + 1 * 210 which is equal to 770 calories. 3 hamburgers and 2 medium cokes is 3 * 280 + 2 * 210 which is equal to 840 + 420 which is equal to 1260 calories. numbers check out so the answer is good.
 Travel_Word_Problems/673742: Jerome rides his bike 10 kilometers everyday. How many meters does he ride? 1 solutions Answer 418826 by Theo(3458)   on 2012-10-30 11:49:41 (Show Source): You can put this solution on YOUR website!there are 1,000 meters in a kilometer so Jerome rides his bike 10,000 meters every day.
 test/673717: Divide using synthetic division. (3x^4+11x^3-20x^2+7x+35) divided by (x+5) 1 solutions Answer 418822 by Theo(3458)   on 2012-10-30 11:48:13 (Show Source): You can put this solution on YOUR website!if you don't know how to do it, then read this tutorial. http://www.purplemath.com/modules/synthdiv.htm it will explain how it's done. if you follow their guidelines, you will get something that looks like this: your divisor of x + 5 becomes the multiplier of -5 because when you set x + 5 = 0 and solve for x, you get x = -5. you bring down the coefficients of all the terms. you then bring down the first coefficient of 3. you then multiply 3 by -5 to get -15 and add that to 11 to get - 4 you then multiply -4 by -5 to get + 20 and add that to - 20 to get 0 you then multiply 0 by -5 to get + 0 and add that to + 7 to get + 7 you then multiply + 7 by - 5 to get - 35 and add that to + 35 to get 0 your anser is 3x^3 - 4x^2 + 7 with a remainder of 0. check out the picture and look at the tutorial and you should be able to figure out how it was done.
 logarithm/671116: what is 346 in base 8 minus 405 in base 8?1 solutions Answer 417272 by Theo(3458)   on 2012-10-24 06:41:56 (Show Source): You can put this solution on YOUR website!good question. what is 346 in base 8 minus 405 in base 8. assuming these numbers are already in base 8, then you would subtract 346 from 405 and then give the answer the negative sign because the number you are subtracting is larger than the number you are subtracting from. you would get: 346 - 405 is equal to - (405 - 346) you would try to subtract the 6 from the 5 which you can't do because the 6 is larger than the 5. you need to borrow from the next position. since the next digit over is 0, you need to go over 1 more digit and borrow from the 4. the 4 becomes 3. the 0 becomes 8 (remember you're in base of 8 not base of 10). now you can borrow from the second digit, so the second digit becomes 7 and the first digit becomes 5 + 8 = 13 you can now subtract 6 from 13 to get 7 for the first digit you can now subtract 4 from 7 to get 3 for the second digit. you can now subtract 3 from 3 to get 0 for the first digit. the answer should be -37 in the base of 8. a good way to check is to convert everything to base of 10, perform the operation, and then convert everything back to the base of 8. to convert 405 from the base of 8 to the base of 10, do the following: 4 * 8^2 + 0 * 8^1 + 5 * 8^0 = 261 405 in the base of 8 is equivalent to 261 in the base of 10. to convert 346 from the base of 8 to the base of 10, do the following: 3 * 8^2 + 4 * 8^1 + 6 * 8^0 = 192 + 32 + 6 = 230. 346 in the base of 8 is equivalent to 230 in the base of 10. now subtract 261 from 230 to get an answer of -31 in the base of 10. now convert -31 from the base of 10 to the base of 8. the way to convert from the base of 10 to the base of 8 is to repeatedly divide the number by 8 and then use the remainders in reverse order from how they were created. as an example, since we already know that 405 in the base of 8 is equal to 261 in the base of 10, we will convert 261 from the base of 10 to the equivalent number in the base of 8. if we get 405 then we know we did it right. 261 divided by 8 = 32 with a remainder of 5 32 / 8 = 4 with a remainder of 0 4 / 8 = 0 with a remainder of 4 going with the remainders in reverse order of ho they were created, you get 405 which is the the number 261 in base 8 format. i'll convert 230 in the base of 10 to its equivalent number in the base of 8. if i get 346, then i did it right. 230 / 8 = 28 with a remainder of 6 28 / 8 = 3 with a remainder of 4 3 / 8 = 0 with a remainder of 3 gathering my remainders in the reverse order from which they created, i get 346 as the equivalent number in the base of 8. since this agrees with what i already knew, then i did it correctly. now back to the original problem. in base of 10, the difference was -31 convert this to octal format as follows: 31 / 8 = 3 with a remainder of 7 3 / 8 = 0 with a remainder of 3 the number in octal format is -37. this agrees with the arithmetic up front so we're good. i got the same answer both ways. way 1 was performing the arithmetic directly in the base of 8. way 2 was converting everything to the base of 10 and performing the arithmetic and then converting the answer back to the base of 8.
 Probability-and-statistics/671115: An exam paper has 6 questions. Each question will be marked by a different person. The markers will be randomly chosen from 11 academic staff, of whom 7 are women. One of the staff members is a professor. Find the probability that the professor is chosen.1 solutions Answer 417271 by Theo(3458)   on 2012-10-24 06:16:09 (Show Source): You can put this solution on YOUR website!here's the deal. you have 11 staff members there is 1 professor among the group. the fact that 7 of the staff members are women is irrelevant to the problem therefore this is extraneous information that you can ignore. i'll solve it 2 ways. both ways will get the same answer which confirms that the answer has a high probability of being right. combination formula used is C(n,x) where n is the total possible choices you have and x is the total possible choices that you want. the formula for C(n,x) is n! divided by (x! * (n-x)!). if you have a good scientific calculator that does combination formulas for you, you can also use that. anyway here goes with method 1. the probability that you will get the professor for one of the 6 questions is equal to 1 minus the probability that you will not get the professor for one of the 6 questions. that probability is equal to 10/11 * 9/10 * 8/9 * 7/8 * 6/7 * 5/6 which is equal to (10! / 4!) / (11! / 5!) which is equal to .45454545..... 1 minus .45454545..... is equal to .54545454..... that's the probability using method 1. method 2 is done as follows: this method uses the combination formula. the solution is the number of ways you can get the professor divided by the number of ways you can get anybody. the number of ways you can get anybody is equal to C(11,6) which is the number of ways you can get any of the 11 staff members to be on a team of 6. C(11,6) is equal to 11! / (6! * 5!) which is equal to 462. you need one professor and you have 1 to choose from so the number of ways you can get the professor is 1C1 = 1. you need 5 other members on the team of 6 that are not professors and you have 10 to choose from so the number of ways you can get the additional 5 members that are not professors is C(10,5) = 252. the total number of ways you can get the professor and the non-professors to be on the team of 6 is therefore equal to 1 * 252 = 252 the number of ways of getting the professor divided by the number of ways of getting anybody is therefore equal to 252/462 = .54545454.... obviously method 1 was easier in this problem. that may not be as true in other problems where you can't just get 1 minus the probability of not getting the professor. both methods match so either method will get you the solution. the second method is trickier but instructive because sometime you are taught to do it this way and it helps to understand how to do it.
 Numbers_Word_Problems/670969: THE PRODUCT OF TWO NUMBERS IS 17. WHAT ARE THE TWO NUMBERS1 solutions Answer 417226 by Theo(3458)   on 2012-10-23 20:28:14 (Show Source): You can put this solution on YOUR website!since 17 is a prime number, one of the numbers has to be 1 and the other number has to be 17.
 Geometry_Word_Problems/670937: a 8 ft board is to be cut into three pieces, two equal length ones and the third 3 in shorter than the other two. If the cutting does not result in any length being lost, how long are the pieces?1 solutions Answer 417223 by Theo(3458)   on 2012-10-23 20:26:13 (Show Source): You can put this solution on YOUR website!the 3d length has to be less than the other 2. this means that the maximum length of the 3d piece has to less than 8/3 feet. if we're assuming that there has to be something left over, then the minimum length of the 3d piece has to be greater than 0. so we have a minimum length of the 3d piece of greater than 0 and a maximum length of the 3d piece of smaller than 8/3. anything in between should be ok. example: let the 3d piece be equal to 1 foot. this means the 2 other pieces are 7/2 feet each. let the 3d piece be equal to 2 feet. this means the 2 other pieces at 6/2 feet each. as long as you don't need to have integers, there are an infinite number of lengths that would satisfy the requirements. if you assume some discrete amount, like no length could have an measurement that doesn't end in an inch, then you now have a finite number of possible solutions. in fact, this probably means that the 3d piece needs to be in 2 inch increments which will allow the 2 equal pieces to be in 1 inch increments. example: 3d piece equals 1 foot which means the 2 equal pieces equal 7/2 = 3.5 feet each. 3.5 is equivalent to 3 and 6 inches which is in increments of an inch so we're good. if we increase 3d section to 1foot 1 inch then we have 6 feet 11 inches left which we divide by 2 to get 3 feet 5.5 inches which is less than inch increments so we're not good. we would need to increase 3d section to 1 foot 2 inches (increments of 2 inches) to get 1 inch increments on the equal segments. 1 foot 2 inches from 8 feet leaves 6 feet 10 inches which divided by 2 equals 3 feet 5 inches for each of the equal segments. not exactly sure what you were looking for but this is the solution based on my understanding of what you are asking.
 Functions/670883: Use the Laws of Logarithms to expand the expression. ln sqrt 9r^8s.... instead of a regular sqrt its a cubed root but its not a 3 its 4.Its the sqrrot sigh with a little 41 solutions Answer 417210 by Theo(3458)   on 2012-10-23 19:49:37 (Show Source): You can put this solution on YOUR website!i believe you are looking for the natural log of root4(9r^8s) root4 is equivalent to taking the fourth root of which is equivalent to raising to the 1/4 power, so we get: ln ((9r^8s)^(1/4)) this becomes: (1/4) * ln ((9r^8s) which becomes: (1/4) * [ ln(9) + ln(r^8s) ] which becomes: (1/4) * ln(9) + (1/4) * ln(r^8s) which becomes: (1/4) * ln(9) + (1/4) * 8 * s * ln(r) to confirm this was done correctly, i let r = 2 and s = 3 and i solved using both the original expression and the final expression and got the same answer of 4.70819 rounded to 5 decimal places. the concepts you use are: log(a^b) = b*log(a) log((a^b)^c) = c*log(a^b) = b*c*log(a) log(a*b) = log(a) + log(b)
 Exponential-and-logarithmic-functions/670878: what is the solution to the equation log4 of 4x + 2log4 of x = 4a 1 solutions Answer 417172 by Theo(3458)   on 2012-10-23 18:52:27 (Show Source): You can put this solution on YOUR website!the answer is x = 4 here's how. your equation is: log4(4x) + 2log4(x) = 4 since a*log(b) = log(b^a), this equation becomes: log4(4x) + log4(x^2) = 4 since log(a) + log(b) = log(a*b), this equation becomes: log4(4x * x^2) = 4 simplify this to get: log4(4x^3) = 4 since logb(x) = y if and only if b^y = x, this equation becomes: 4^4 = 4x^3 since 4^4 = 256, this equation becomes: 256 = 4x^3 divide both sides of this equation by 4 to get: 64 = x^3 take the third root of both sides of this equation to get: x = 4
 Word_Problems_With_Coins/670863: I am not sure how to figure out a coin problem without how much it's worth for example: A bank contains 44 coins (nickles, dimes, quarters). There are twice as many dimes as nickles and 8 fewer nickles than quarters. How much money is in the bank?1 solutions Answer 417165 by Theo(3458)   on 2012-10-23 18:41:48 (Show Source): You can put this solution on YOUR website!44 coins in total. n = number of nickels. d = number of dimes q = number of quarters. there are twice as many dimes as nickels. d = 2n 8 fewer nickels than quarters. n = q - 8 how much money is in the bank. the amount of money in the bank is determined as follows: .05n = amount of money in nickels. .1d = amount of money in dimes. .25q = amount of money in quarters. you have 2 equations. n + d + q = 44 .05n + .1d + .25q = amount of money in the bank. you need to solve for n, d, and q and then you can determine how much money is in the bank. start with the equation of n + d + q = 44 you are given that d = 2n so you can substitute 2n for d which makes the equation becomes: n + 2n + q = 44 you are given that n = q - 8 so you can substitute q - 8 for n which makes the equation become: q - 8 + 2(q - 8) + q = 44 since you now have 1 unknown in 1 equation, you can solve for q. simplify the equation to get: q - 8 + 2q - 16 + q = 44 combine like terms to get: 4q - 24 = 44 add 24 to both sides of the equation to get: 4q = 44 + 24 = 68 divide both sides of the equation by 4 to get: q = 68/4 = 17 since you know that n = q - 8, this means that n = 9 you have: q = 17 n = 9 since you know that d = 2n, this means that d = 18 now you have: q = 17 n = 9 d = 18 the first equation you had to solve is n + d + q = 44 replace q with 17 and n with 9 and d with 18 and you get: 9 + 18 + 17 = 44 which becomes 44 = 44 which is true so the numbers for n and d and q look good. now to the money. the equation is: .05n + .1d + .25q = the money replace n with 9 and d with 18 and q with 17 and you get: .05(9) + .1(18) + .25(17) = .45 + 1.8 + 4.25 which is equal to 6.5 dollars that you have in the bank.
 Rational-functions/670856: In triangle DEF, the equation of the altitude from D through EF is 4x + 3y - 13 = 0 and the equation of the altitude from E through DF is 2x - y - 4 = 0. What do you call the point where these lines intersect? Find the point where these lines intersect. Explain each step of your solution. 1 solutions Answer 417155 by Theo(3458)   on 2012-10-23 18:21:18 (Show Source): You can put this solution on YOUR website!basically you're talking about the intersection of 2 lines whose equations are: 4x + 3y - 13 = 0 2x - y - 4 = 0 the fact that these lines are altitudes to sides of a triangle doesn't change the fact that you are simply looking for the point of their intersection. if there was additional information they were looking for, it would make a difference, but the way the problem is structured, all that other information is extraneous to the heart of the problem which is to find the intersection of the 2 lines. the 2 equations are, once again: 4x + 3y - 13 = 0 2x - y - 4 = 0 you need to solve these equations simultaneously to get the solution. you need to first transform them to standard form which is ax + by = c 4x + 3y - 13 = 0 becomes 4x + 3y = 13 once you convert it. 2x - y - 4 = 0 becomes 2x - y = 4 once you convert it. your 2 equations are now: 4x + 3y = 13 2x - y = 4 we will solve by elimination. multiply the second equation by 2 to get: 4x - 2y = 8 your 2 equations are now: 4x + 3y = 13 4x - 2y = 8 subtract the second equation from the first equation to get: 5y = 5 divide both sides of this equation by 5 to get: y = 1 substitute for y in either original equation to find the value of x. we'll use: 4x + 3y = 13 replace y with 1 to get: 4x + 3 = 13 subtract 3 from both sides of this equation to get: 4x = 10 divide both sides of this equation by 4 to get: x = 2.5 your solution should be: x = 2.5 y = 1 this solution should apply to both equation. the first original equation is: 4x + 3y = 13 replace x with 2.5 and y with 1 to gtet: 4(2.5) + 3(1) = 13 which becomes: 10 + 3 = 13 which is true. the second original equation is: 2x - y = 4 replace x with 2.5 and y with 1 to get: 2(2.5) - 1) = 4 which becomes: 5 - 1 = 4 which is also true. the solution applies to both equations and is confirmed as good. the point where these lines intersect is (x,y) = (2.5,1). that intersection point is the solution that is common to both equations.
 Evaluation_Word_Problems/670820: Many drugs used to treat cancer are expensive. BusinessWeek reported on the cost per treatment of Herceptin, a drug used to treat breast cancer (BusinessWeek, January 30, 2006). Typical treatment costs (in dollars) for Herceptin are provided by a simple random sample of 10 patients. 4,357 5,564 2,698 4,906 4,482 4,804 6,449 4,108 4,229 3,809 Develop a point estimate of the standard deviation of the cost per treatment with Herceptin (to 1 decimal). 1 solutions Answer 417151 by Theo(3458)   on 2012-10-23 18:09:09 (Show Source): You can put this solution on YOUR website!the mean of this data is equal to 4,541 the standard deviation is equal to 1008. the point estimate of this data is the mean. at 95% confidence level, the range of possible values you can expect in a sample of the same size 95% of the time would be equal to +/- 1.96 standard deviations about the mean which would be calculated as 4541 - 1008 * 1.96 to 4541 + 1008 * 1.96. that would make the interval estimate from 2565.3 to 6516.7 the point estimate is 4541. the standard deviation is 1008.
 Mixture_Word_Problems/670719: You need a(n) 2% acid solution, and you'll get it by adding a(n) 10% acid solution to 25 liters of a(n) 1% acid solution. How many liters must be added? The amount of the 2% solution to be added is ? liters?? You need a(n) 5% acid solution, which you'll get by adding a(n) 16% acid solution to 21 liters of a(n) 3% acid solution. How many liters must be added? The amount of the 16% solution to be added is ? liters.1 solutions Answer 417106 by Theo(3458)   on 2012-10-23 16:51:31 (Show Source): You can put this solution on YOUR website!x + y = z x number of liters of 10% solution y = number of liters of 1% solution. z = number of liters of 2% solution. .10x + .01y = .02z the 2 equations you need to deal with are: x + y = z (number of liters of each solution) .10x + .01y = .02z (number of liters of acid in each solution). you know that y = 25 because that's given. your formulas becomes: x + 25 = z .10x + .01(25) = .02z from the first equation, solve for z to get z = x + 25 replace z with x + 25 in the second solution to get: .10x + .01(25) = .02(x + 25) solve for x in this equation. simplify the equaton to get: .10x + .25 = .02x + .5 subtract .02x from both sides of this equation and subtract .25 from both sides of this equation to get: .10x - .02x = .5 - .25 simplify to get: .08x = .25 divide both sides of this equation by .08 to get: x = .25 / .08 = 3.125 you need to add 3.125 liters of a 10% solution to 25 liters of a 1% solution to get a 2% solution. since y = 25 and x = 3.125, then z must be equal to 28.125 x + y = z becomes 3.125 + 25 = 28.125 .10x + .01y = .02z becomes: .10(3.125) + .01(25) = .02(28.125) simplify this to get: .3125 + .25 = .5625 which becomes: .5625 = .5625 numbers check out so the solution is good. you get .5625 liters of acid in the combined solution which is 2% of 28.125 liters.