Algebra ->  Tutoring on algebra.com -> See tutors' answers!      Log On

 Tutoring Home For Students Tools for Tutors Our Tutors Register Recently Solved
 By Tutor
| By Problem Number |

Tutor:

# Recent problems solved by 'Theo'

Jump to solutions: 0..29 , 30..59 , 60..89 , 90..119 , 120..149 , 150..179 , 180..209 , 210..239 , 240..269 , 270..299 , 300..329 , 330..359 , 360..389 , 390..419 , 420..449 , 450..479 , 480..509 , 510..539 , 540..569 , 570..599 , 600..629 , 630..659 , 660..689 , 690..719 , 720..749 , 750..779 , 780..809 , 810..839 , 840..869 , 870..899 , 900..929 , 930..959 , 960..989 , 990..1019 , 1020..1049 , 1050..1079 , 1080..1109 , 1110..1139 , 1140..1169 , 1170..1199 , 1200..1229 , 1230..1259 , 1260..1289 , 1290..1319 , 1320..1349 , 1350..1379 , 1380..1409 , 1410..1439 , 1440..1469 , 1470..1499 , 1500..1529 , 1530..1559 , 1560..1589 , 1590..1619 , 1620..1649 , 1650..1679 , 1680..1709 , 1710..1739 , 1740..1769 , 1770..1799 , 1800..1829 , 1830..1859 , 1860..1889 , 1890..1919 , 1920..1949 , 1950..1979 , 1980..2009 , 2010..2039 , 2040..2069 , 2070..2099 , 2100..2129 , 2130..2159 , 2160..2189 , 2190..2219 , 2220..2249 , 2250..2279 , 2280..2309 , 2310..2339 , 2340..2369 , 2370..2399 , 2400..2429 , 2430..2459 , 2460..2489 , 2490..2519 , 2520..2549 , 2550..2579 , 2580..2609 , 2610..2639 , 2640..2669 , 2670..2699 , 2700..2729 , 2730..2759 , 2760..2789 , 2790..2819 , 2820..2849 , 2850..2879 , 2880..2909 , 2910..2939 , 2940..2969 , 2970..2999 , 3000..3029 , 3030..3059 , 3060..3089 , 3090..3119 , 3120..3149 , 3150..3179 , 3180..3209 , 3210..3239 , 3240..3269 , 3270..3299 , 3300..3329 , 3330..3359 , 3360..3389 , 3390..3419 , 3420..3449 , 3450..3479, >>Next

 Trigonometry-basics/612804: A function f is defined by f(x)= cos(^2)x + 1/csc2x. What is f(pi/6)?1 solutions Answer 385726 by Theo(3464)   on 2012-05-20 05:16:59 (Show Source): You can put this solution on YOUR website!function is defined as: f(x) = cos^2(x) + 1/csc(2x) what is f(pi/6) you replace x with pi/6 and solve. cos^2(pi/6) = [cos(pi/y)]^2= (.866025)^2 = .75 1/csc(2x) = sin(2x) = sin(2pi/6) = .8660254038 cos^2(pi/6) + 1/csc(2x) = .75 + .86... = 1.616025404 if you convert this to degrees it might be easier to see. you convert pi/6 to degrees by multiplying it by 180 and dividing it by pi. pi/6 * 180/pi = 180/6 = 30 degrees. cosine 30 degrees = sqrt(3)/2 sine 60 degrees = sqrt(3)/2 cos^2(30) = (sqrt(3)/2)^2 = 3/4 1/csc(2x) = sin(2x) because sine = 1 / cosecant and cosecant = 1 / sine. x = 30 so 2x = 60 degrees. sine(60) = sqrt(3)/2 cos^2(30) + sin(60) = 3/4 + sqrt(3)/2 = (3 + 2*sqrt(3))/4 = 1.616025404. this is the same answer we got before using radians. the graph of the equation of cos^2(x) + 1/csc(2x) is shown below. getting it to graph properly is a challenge, but if you know a few tricks, it can be done. first trick: cos^2(x) is the same as (cos(x))^2 first you find cos(x) and then you square it. second trick: csc(2x) = 1/sin(2x) and sin(2x) = 1/csc(2x) by definition. sin(2x) is the sine of 2 times the angle. if the angle is 30 degrees, then 2x is 60 degrees. last, but not least, is that the graphing software deals in radians and not degrees. on the graphing software, the value of x = pi/6 radians will show up as .523598776 because pi/6 is equal to 3.141592654... / 6 which is equal to .5235..... if you're using a scientific calculator, make sure to set it to radians before trying to solve this problem directly. if you want to work with degrees, make sure to convert the radians to degrees before working the problem.
 Distributive-associative-commutative-properties/612769: what is the positive x-intercept of the quadratic function f(x)=x^2+2x-631 solutions Answer 385724 by Theo(3464)   on 2012-05-20 04:39:46 (Show Source): You can put this solution on YOUR website!should be x = 7 factor the quadratic equation as follows: (x-a)*(x+b) = 0 you want an a and a b such that b-a = 2 look at the factors of a*b that equal 63. you get: 63*1 21*3 7*9 looks like 7 and 9 will do it because 9-7 = 2 set b = 9 and a = 7 and you get: (x-7) * (x+9) = 0 results in x = 7 or x = -9 positive x-intercept is 7. graph of the equation of x^2 + 2x - 63 confirms it as shown below:
 Surface-area/592456: How do I find the area of a right triangle where it gives me one leg and one angle in adition to the right? Say,the leg given is 16, opposite an angle 37 degrees? How do I find the other leg and hypotonuse, then finding the area?1 solutions Answer 375961 by Theo(3464)   on 2012-03-27 19:48:24 (Show Source): You can put this solution on YOUR website!you have a right triangle. one angle other than the right angle. one side opposite the angle. you can use the sine function to get the hypotenuse or you can use the tan function to get the side opposite the other angle. sine (37) = 16 / hypotenuse multiply both sides of this equation by the hypotenuse and divide both sides of this equation by sine (37) to get: hypotenuse = 16 / sine(37) use your calculator to find the sine(37) and then solve for the hypotenuse. similarly. tangent (37) = 16 / side opposite the other angle. multiply both sides of the equation side opposite the other angle and divide both sides of the equation by tangent (37) to get: side opposite the other angle = 16 / tangent (37). if you do this correctly, you will get: hypotenuse = 26.58624226 side opposite the other angle = 21.23271715 to confirm you did this correctly, use the pythagorean formula to get: (16)^2 + (21.23271715)^2 = (26.58624226)^2 this results in: 706.0282774 = 706.8282774 confirming the values for the hypotenuse and the other leg are good.
 �t�e�s�t/591583: Hello I did this question in my maths test today and I did not get it.here it is please help. this a table X y=2x-3 -3 ? -2 ? -1 ? 0 ? 1 ? 2 ? 3 ? There is a Line between the x coloumn and the y= 2x-3 coloumn can you please fill in the question marks and tell me the answer as well and explain it please and I think you are suppose to find the y coordinates or point I think using the left side of the table and there is another question related to this one above. You will have to draw a graph for this one x axis= 5 to -5 going left from 5. y axis from 5 to -5 going down from 5 Thanks please help I have missed 2 weeks of school because of a broken arm which i had to get an operation on. 1 solutions Answer 375680 by Theo(3464)   on 2012-03-26 06:15:15 (Show Source): You can put this solution on YOUR website!a graph of the equation of y = 2x-3 is shown below: your table will look like this: ```x y = 2x-3 -3 -9 -2 -7 -1 -5 0 -3 1 -1 2 1 3 3 ``` all you're doing is replace x in the equation of y = 2x-3 with the value of x. when x = -3, y = 2x-3 becomes y = 2(-3) - 3 which becomes y = -6 - 3 which becomes y = -9 similarly, when x = 3, y = 2x-3 becomes y = 2(3) - 3 which becomes y = 6 - 3 which becomes y = 3. etc.
 Linear-systems/591614: If I walk 50kms. The first day I walk so far the next day I walk half as much and on the 3rd day I walk a third of the day before and on the fourth day a fourth of the day previously, How far did I walk the first day1 solutions Answer 375678 by Theo(3464)   on 2012-03-26 05:38:40 (Show Source): You can put this solution on YOUR website!you don't have enough information to come up with a single answer. you can solve this problem, but you will need to know either: total time or: total distance the basic formula is RT = D R = rate of travel T = time D = distance i don't know what you mean by kms. if that's kilometers per second then it's not realistic. the average person walks 3 miles per hour which is roughly equivalent to 5 kilometers per hour. that would be 5kph. i will assume 5kph. assuming that, this is how it will work out. the distance traveled is shown as follows: day 1 equals D day 2 equals D/2 day 3 equals (D/2)/3 which is equal to D/6 day 4 equals (D/6)/4 which is equal to D/24 the total distance traveled over the 4 days is: D + D/2 + D/6 + D/24 this comes out to be 41D/24 total. D = 24D/24 D/2 = 12D/24 D/6 = 4D/24 D/24 = D/24 since they now have a common denominator, just add up the numerators and divide the total by the denominator to get (24 + 12 + 4 + 1)/24 which is equal to: 41D/24 the rate is always the same (5kmph) the total time is calculated by use of the formula to get: total time * rate = total distance we solve for time to get: total time = total distance / rate since total distance is equal to 41D/24, and since rate is equal to 5kmph, this becomes: total time = (41D/24) / 5 which becomes 41D/(24*5) which becomes .3416666667D your final equation is:" total time = .3416666667D you have 2 unknowns in one equation. best you can do is make an assumption about total time or about total distance. when you know one of them, you can find the other. you can't, however find both. not with the information given. i assumed the total time was 4 hours. the equation of: total time = .3416666667D becomes: 4 = .3416666667D i then solved for D to get: D = 4/.3416666667 = 11.70731707 kilometers that distance only applies if the total time is 4 hours. change the total time and the distance changes. assuming that the total time is 4 hours, the day by day breakdown was: ```day distance (km) rate (kmph) time (hours) 1 11.70731707 5 2.341463415 2 5.853658537 5 1.170731707 3 1.951219512 5 0.3902439024 4 .487804878 5 0.0975609756 total 20 --- 4 ``` total distance was 20 kilometers and total time was 4 hours. distance traveled the first day was 11.70731707 kilometers this value is only accurate if the total time was assumed to be 4 hours. once again, there is not enough information provided to give an exact answer. you have too many unknowns and not enough equations. to get 1 answer, you need to have as many equations as you have unknowns (variables). you had 2 unknowns (time and distance) with only 1 equation (rate * time = distance).
 test/591605: (3x^3y^-2)(-2xy^3)^-11 solutions Answer 375677 by Theo(3464)   on 2012-03-26 04:45:24 (Show Source): You can put this solution on YOUR website!(3x^3y^-2)(-2xy^3)^-1 is your expression. since, in general, x^-a = 1/x^a, this expression can become: (3x^3y^-2) / (-2xy^3) since, in general, (a*b)/(c*d) is the same as (a/c) * (b/d), this expression can become: -(3/2) * (x^3/x) * (y^-2/y^3) since, in general, x^a / x^b is equal to x^(a-b), this expression can become: -(3/2) * (x^3-1) * (y(-2-3) which can be simplified to: -(3/2) * x^2 * y^-5 since, in general, x^-a = 1/x^a, this expression can become: -(3/2) * x^2 * 1/y^5 since, in general, (a/c) * (b/d) is the same as (a*b) / (c*d), this expression can become: (-3*x^2*1) / (2*y^5) which can be simplified to: (-3x^2)/(2y^5) that's your solution. you can confirm the solution is good by picking value for x and for y and using those values in the original equation and the final equation. if the solution comes out the same, then you did good. you can use your calculator to perform the calculations. i did with the values of x = 5 and y = 7. i got: original equation equals: .0022312132 final equation equals: .0022312132 they're the same so the original equation is equivalent to the final equation so we're good.
 testmodule/591606: log4(x+2)-log4(x-1)=11 solutions Answer 375676 by Theo(3464)   on 2012-03-26 04:21:21 (Show Source): You can put this solution on YOUR website!log(4,x+2) - log(4,x-1) = 1 since, in general, log(x) - log(y) = log(x/y), this equation becomes: log(4,(x+2)/(x-1)) = 1 since, in general, log(b,x) = y if and only if b^y = x, this equation becomes: 4^1 = (x+2)/(x-1) which is the same as: (x+2)/(x-1) = 4 multiply both sides of this equation by (x-1) to get: (x+2) = 4*(x-1) simplify to get: x+2 = 4x-4 subtract x from both sides of this equation and add 4 to both sides of this equation to get: 6 = 3x which is the same as: 3x = 6 divide both sides of this equation by 3 to get: x = 2 replace x with 2 in your original equation. log(4,x+2) - log(4,x-1) = 1 becomes: log(4,(4) - log(4,1) = 1 this is true if and only if 4^1 = 4 since 4^1 is equal to 4, this confirms the solution of x = 2 is good. you can also confirm it by solving it using the LOG function of your calculator. to do that, you need to convert the equation as follows: log(4,(4) - log(4,1) = 1 becomes: LOG(4)/LOG(4) - LOG(1)/LOG(4) = 1 use the log function of your calculator to solve to get: you get: 1 - 0 = 1 which becomes 1 = 1 which is true, confirming x = 2 is good. LOG(4) / LOG(4) is equivalent to log(10,4) / log(10,4) LOG(1) / LOG(1) is equivalent to log(10,1) / log(10,4) since, in general, x/x = 1, LOG(4)/LOG(4) = 1 since log(10,1) = y if and only if 10^y = 1, this occurs only when y = 0 which makes the expression log(10,1) equivalent to 0 which is confirmed when you use your calculator to solve for LOG(1). you get 0 as the answer.
 Miscellaneous_Word_Problems/590731: There are three different flavors of juice in a cooler.There is the same number of bottles of apple juice as grapefruit juice. There are four more bottles of orange juice than the other kinds combined. How many bottles of orange juice are there in the cooler? 1 solutions Answer 375269 by Theo(3464)   on 2012-03-23 15:24:17 (Show Source): You can put this solution on YOUR website!T = total number of bottles in the cooler. x = number of bottles of grapefruit juice. x = number of bottles of apple juice. 2x + 4 = number of bottles of orange juice. T = x + x + 2x + 4 combine like terms to get: T = 4x + 4 number of bottle of orange juice is equal to 4x + 4 - 2x the number of bottles of orange juice is equal to 2x + 4. you don't have enough information to narrow it down better than that. you either need to know the total number of bottles or the number of bottles of apple or graph juice.
 test/590789: A \$4000 loan, taken now, with a simple interest rate of 8%, per year, will require a total repayment of \$4640. When will the loan mature?1 solutions Answer 375268 by Theo(3464)   on 2012-03-23 15:16:43 (Show Source): You can put this solution on YOUR website!interest rate is 8% per year = .08 per year total repayment is 4640 on a loan of 4000. total interest to be paid is 640. total interest each year is calculated at .08 * 4000 = 320 per year. 640 / 320 = 2 years when the total amount on the loan equals 4640 which is when the loan matures. paying the loan before 2 years is up requires less interest to be paid. paying the loan after 2 years is up requires more interest to be paid. total due is 4640 in exactly 2 yeara.
 Expressions-with-variables/590798: I am having a terrible time doing word problems in algebra, simple for some but not for me. Could someone please help with two problems, thank you very much. A university bookstore recently sold a wire bound graph-paper notebook for \$2.50, and a college-ruled notebook for \$2.30. At the start of spring semester, a combination of 50 of these notebooks were sold for a total of \$118.60. How many of each type were sold? A bricklayer and an electrician together spend 90 hours working on a new house. If the bricklayer earns \$12 per hour, the electrician earns \$16 an hour, and the owner pays them a total of \$1350 for their work, how many hours does each worker spend on the house? 1 solutions Answer 375266 by Theo(3464)   on 2012-03-23 15:12:37 (Show Source): You can put this solution on YOUR website!A university bookstore recently sold a wire bound graph-paper notebook for \$2.50, and a college-ruled notebook for \$2.30. At the start of spring semester, a combination of 50 of these notebooks were sold for a total of \$118.60. How many of each type were sold? w = number of wire bound notebooks. c = number of college ruled notebooks. total of 50 notebooks were sold for \$118.60. your equations are: 2.50w + 2.30c = 118.60 w + c = 50 these are 2 equations that need to be solved simultaneously. you can use substitution. from the second equation, solve for w to get: w = 50 - c substitute for w in the first equation to get: 2.50*(50-c) + 2.30c = 118.60 simplify to get: 125 - 2.50c + 2.30c = 118.60 combine like terms to get: 125 - .20c = 118.60 subtract 125 from both sides of the equation to get: .20c = -6.4 divide both sides of the equation by -.20 to get: c = -6.4/-.20 = 32 since w + c = 50, this means that w equals 18. you have w = 18 and c = 32 w + c = 18 + 32 = 50 2.50*w + 2.30*c = 118.60 becomes: 2.50*18 + 2.30*32 = 118.60 answer is good. 18 wire bound notebooks are sold 32 college ruled notebooks are sold. A bricklayer and an electrician together spend 90 hours working on a new house. If the bricklayer earns \$12 per hour, the electrician earns \$16 an hour, and the owner pays them a total of \$1350 for their work, how many hours does each worker spend on the house? x = number of hours the bricklayer works y = number of hours the electrician works x + y = 90 12x + 16y = 1350 you have 2 equations that need to be solved simultaneously. you can solve by substitution. solve for y in the first equation to get: y = 90 - x substitute for y in the second equation to get: 12x + 16*(90-x) = 1350 simplify to get: 12X + 1440 - 16X = 1350 combine like terms to get: -4x + 1440 = 1350 subtract 1440 from both sides of the equation to get: -4x = 1350 - 1440 = -90 divide both sides of the equation by -4 to get: x = 22.5 since x + y = 90, this means that y is equal to 90 - 22.5 = 67.5 x = 22.5 y = 67.5 x + y = 90 12*x + 16*y = 1350 12*22.5 + 16*67.5 = 270 + 1080 = 1350 answers are good. bricklayer works 22.5 hours electrician works 67.5 hours
 Surface-area/590802: Help I'm confused about this question. How do I solve it?: The height of an equilateral triangular prism increases by three units. The new volume is more than the original by how much? If I were to times it by three, the area of the base, would the answer be in word form: three times the area of the base? 1 solutions Answer 375265 by Theo(3464)   on 2012-03-23 14:57:09 (Show Source): You can put this solution on YOUR website!the area of the base remains the same. that appears to be a given. only the height has changed. the volume of a prism is equal to the height * the area of the base. assume the area of the base is y and assume the height is x, then the original volume is equal to x*y. now the height is increased by 3 units. that makes the height equal to (x+3). the new volume is now equal to (x+3)*y which becomes xy + 3y the new volume is xy + 3y the old volume is xy it appears that the new volume is more than the original volume by 3 times the area of the base. i believe that's your answer and it would be worded that way. 3 times the area of the base is the additional volume measured in cubic units. the area of the base is measured in square units.
 Trigonometry-basics/590664: how do i shift f(x) = 4cos(3x)+1 to the right by pi/12?1 solutions Answer 375231 by Theo(3464)   on 2012-03-23 06:07:44 (Show Source): You can put this solution on YOUR website!your equation is: f(x) = 4cos(3x)+1 you want to shift it to the right by pi/12. this means that if x is equal to pi/12, you want to get the cosine of x = 0. the result of that is that the cosine of 0 is now the cosine of pi/12 which means that value has been shifted to the right by pi/12. the general formula for a trigonometric function is: y = a*cos(b(x-c))+d a is the amplitude. b is the frequency c is the horizontal shift d is the vertical shift of the center line. your equation has an amplitude of 4 and a vertical shift of 0. your equation has a frequency of 3 which means the period is equal to 2*pi/3 which makes it equal to (2/3)*pi instead of the 2*pi that it normally is. based on what you have, the standard form of your equation would be: f(x) = 4*cos(3*(x-0))+1 a graph of your equation looks like this: a vertical line is shown at x = pi/12 to show you where your shift will occur. since you want to shift the graph to the right, then the value of cosine at x = 0 will then become the value of cosine at x = pi/12. to do this, your equation has to go from: f(x) = 4*cos(3*(x-0))+1 to: f(x) = 4*cos(3*(x-(pi/12)))+1 the graph of that equation looks like this:
 Angles/590678: an angle which lies on the vertex of a circle is called ?1 solutions Answer 375230 by Theo(3464)   on 2012-03-23 05:30:03 (Show Source): You can put this solution on YOUR website!a circle has no vertex. if you mean an angle whose vertex lies on the circumference of a circle, then you are talking about an inscribed angle. a central angle has its vertex at the center of the circle. the degree measure of a central angle is equal to the degree measure of its intercepted arc. the degree measure of an inscribed angle is equal to 1/2 the degree measure of its intercepted arc. the intercepted arc is the part of the circumference of the circle that is between the sides of the angle formed. try this reference to learn more. http://www.mathopenref.com/circleinscribed.html
 Linear-equations/590667: Write the equation in standard form of a line satisfying the given conditions through (-5,9); horizontal1 solutions Answer 375229 by Theo(3464)   on 2012-03-23 05:19:55 (Show Source): You can put this solution on YOUR website!the equation is y = 9 the value of y will always be 9 because the line is horizontal. the value of x can be anything, including x = -5. the point (-5,9) is on the line.
 Linear-equations/590619: write each equation in slope-intercept form. Then determine, without solving the system, whether the system has exactly one solution, no solution, or an infinite number of solutions 4x-3y=2 5x+3y=6 12x-9y=6 10x+6y=21 solutions Answer 375228 by Theo(3464)   on 2012-03-23 05:18:41 (Show Source): You can put this solution on YOUR website!4x-3y=2 5x+3y=6 12x-9y=6 10x+6y=2 y = (4/3)x - 2/3 y = (-5/3)x + 2 y = (4/3)x - 2/3 y = -5/3)x + 1/3 2 of the lines have the same slope and the same y intercept so they are identical. if you graph these equations, they will look like one line rather than 2. 2 of the lines have the same slope and different y intercept so they are parallel. you have 1 set of identical lines and 1 set of parallel lines that will intersect in 2 separate points on a graph. for there to be a common solution, all 4 lines would have had to intersect in one common point on the graph. since there is not one common solution for all 4 equations, this system of equations has no solution. a graph of the lines will show the 2 points of intersection.
 Probability-and-statistics/590674: you have a book collection that consists of 20 horror novels, 15 novels,and 25 mystery novels. you randomly pick 4 books to read during a long trip. what is the probability that you pick at least one book of each type?1 solutions Answer 375227 by Theo(3464)   on 2012-03-23 04:59:19 (Show Source): You can put this solution on YOUR website!a = 20 = number of horror novels. b = 15 = number of regular novels. c = 25 = number of mystery novels. total number of books is equal to 20 + 15 + 25 = 60. you pick 4 books at random. you need to pick at least 1 of each type. 3 things have to happen. you do not pick 0 type a or you do not pick 0 type b or you do not pick 0 type c. the probability of picking 0 type a is equal to 40/60 * 39/59 * 38/58 * 37/57 = P(40,4)/P(60,4). the probability of picking 0 type b is equal to 45/60 * 44/59 * 43/58 * 42/57 = P(45,4)/P(60,4). the probability of picking 0 type c is equal to 35/60 * 34/59 * 33/58 * 32/47 = P(35,4)/P(60,4). the combined probability of picking 0 type a or picking 0 type b or picking 0 type c is equal to: (P(40,4) + P(45,4) + P(35,4)) / (P(60,4) this comes out to be equal to .600336317 this is the probability that you will get {0 type a or 0 type b or 0 type c}. the probability that you will not get {0 type a or 0 type b or 0 type c} is equal to 1 minus .600336317 which is equal to .399663683. not getting 0 of each type means that you will get at least 1 of each type. this is difficult to confirm that the formulas work because the possible combinations are 81 and each one of those possible combinations has its own probability, the sum of which must be equal to 1. it's easier to show if you have 2 choices of books and are drawing 3 books. in this case, there are 8 possible combinations so it's easier to show the probability of each combination and to show that the sum of the probabilities is equal to 1. it's also easier to then confirm that the formula of 1 minus the probability of {0 type a or 0 type b} provides the correct answer. those details are shown below: ```number of type a 40 number of type b 20 total 60 summary probability formula used p(0 type a) = 0.033313852 equals 20/60*19/59*18/58 *** p(1 type a and 2 type b) = 0.222092344 equals 3*40/60*20/59*19/58 p(2 type a and 1 type b) = 0.455873758 equals 3*40/60*39/59*20/58 p(0 type b) = 0.288720047 equals 40/60*39/59*38/58 *** total probability 1 details probability formula used aaa 0.288720047 equals 40/60*39/59*38/58 *** aab 0.151957919 equals 40/60*39/59*20/58 aba 0.151957919 equals 40/60*20/59*39/58 abb 0.074030781 equals 40/60*20/59*19/58 baa 0.151957919 equals 20/60*40/59*39/58 bab 0.074030781 equals 20/60*40/59*19/58 bba 0.074030781 equals 20/60*19/59*40/58 bbb 0.033313852 equals 20/60*19/59*18/58 *** total probability 1 ``` the summary shows you that the probability of getting 0 type a is the same as the probability of getting 3 type b. it also shows you that the probability of getting 0 type b is the same as the probability of getting 3 type a. the probability of getting 3 type a or 3 type b is shown in the details. the probability of getting at least 1 of each type is equal to 1 minus the probability of getting 0 type a or 0 type b. the same procedure is used in the larger numbers where we picked 4 books from a selection of 3. finding the number of 0 type a combination was more complicated in this case because there were 16 combinations where 0 type a were picked and 16 combinations where 0 type b were picked and 16 combinations where 0 type c were picked. for example, the possible combinations where type a books were not picked are: bbbb bbbc bbcb bbcc bcbb bcbc bccb bccc cbbb cbbc cbcb cbcc ccbb ccbc cccb cccc this made it difficult to show the details because the individual calculations became onerous to say the least. it was much easier to reduce the number of possibilities in order to show you how it worked. all of these combinations, however, involved 1 minus the probability of getting 0 type a or 0 type b or 0 type c. the same calculations in the simpler example produced the same result, i.e. 1 minus probability of 0 type a minus probability of 0 type b gave the result of at the probability of least 1 of each type.
 Exponents/590439: Could someone please help me with this problem? 3x^2 + 3x^2 1 solutions Answer 375169 by Theo(3464)   on 2012-03-22 17:48:25 (Show Source): You can put this solution on YOUR website!3x^2 + 3x^2 = (3+3) * x^2 = 6x^2
 Exponential-and-logarithmic-functions/590442: Hello, I was wondering if you could please help me evalute this logarithmic problem. I have a decent knowledge of the subject; however, I can am flustered by this problem. The instructions are "Solve the equation". 1/3(6)^(-4x)+1=6 Thank you for your time.1 solutions Answer 375162 by Theo(3464)   on 2012-03-22 17:39:36 (Show Source): You can put this solution on YOUR website!1/3(6)^(-4x)+1=6 i think your equation is this: if so, then you would solve it as follows: subtract 1 from both sides of the equation and then multiply both sides of the equation by 3 to get: take the log of both sides of this equation to get: since log(x^a) = a*log(x), this equation becomes: divide both sides of this equation by to get: this becomes: -4x = 1.511391594 divide both sides of this by -4 to get: x = -.0377847899
 Triangles/590436: How do I find the height of a triangle when I only have it's base?1 solutions Answer 375158 by Theo(3464)   on 2012-03-22 17:21:46 (Show Source): You can put this solution on YOUR website!you can't. you need more information. 1 side is not enough. 2 sides of the triangle and an included angle would be enough. 3 sides of the triangle would be enough. other combinations might work but 1 side by itself is not enough.
 Sequences-and-series/590429: How do I prove the following sequence is geometric? tn = 24 x (-1/2)n-11 solutions Answer 375156 by Theo(3464)   on 2012-03-22 17:16:29 (Show Source): You can put this solution on YOUR website!if it has a common ratio then it's geometric. if it has a common difference then it's arithmetic. your equation is: Tn = 24 x (-1/2)^(n-1) your common ratio looks like it's (-1/2) let's see how this works. T1 = 24 * (-1/2)^0 = 24 * 1 = 24 T2 = 24 * (-1/2)^1 = 24 * -1/2 = -12 T3 = 24 * (-1/2)^2 = 24 * 1/4 = 6 T4 = 24 * (-1/2)^3 = 24 * -1/8 = -3 if you look at your number sequence you will see that the common ratio is (-1/2) 24 * (-1/2) = -12 * (-1/2) = 6 * (-1/2) = -3, etc.
 Probability-and-statistics/590337: probability of failure of compnent is 0.06. Than what is the probability of failure of 109 components 1 solutions Answer 375155 by Theo(3464)   on 2012-03-22 17:11:47 (Show Source): You can put this solution on YOUR website!if the probability of failure of one component is .06, then the probability of success of one component is 1 - .06 = .97 if you take a sample of n components, then the formula for the number of failures in that sample would be: p(x failures) = C(n,x) * p^x * q^(n-x) C(n,x) equals the combination formula of n! / (x! * (n-x)!) p^x equals the probability of failure raised to the power of x. q^(n-x) equals the probability of success raised to the power of (n-x). example: you draw a sample of 3 components. ```p(0 failures) = C(3,0) * .06^0 * .94^3 which equals .830584 p(1 failure) = C(3,1) * .06^1 * .94^2 which equals .159048 p(2 failures) = C(3,2) * .06^2 * .94^1 which equals .010152 p(3 failures) = C(3,3) * .06^3 * .94^0 which equals .000216 total of all possible probabilities is equal to: 1.00000 ``` the sum of all probabilities is equal to 1 as it should be. if you draw a sample of 1000 components, then the sum of all probabilities will still be equal to 1 but each possible probability will be much less. example: the probability of exactly 106 defective components out of the sample of 1000 components is equal to: C(1000,106) * .06^106 * .97^894 which is equal to ????? my calculator is unable to calculate this. it tells me that C(100,106) is too large for it to handle (overflow error). it tells me that .06^106 is equal to 0. it tells me that .94^894 is equal to 9.468934625 * 10^(-25) suffice it to say that the probability of getting 106 defective components out of a batch of 1000 components is extremely small. try something like 50 failures out of a batch of 100 components. you get: p(50 failures) = C(100,50) * .06^50 * .94^50 which is equal to 3.696656602 * 10^(-34) that's a very small number.
 Systems-of-equations/590332: Help please!! Find the equation of the line of best fit from the data in the table. Round the slope and the y-intercept to the nearest hundredth. x 12 14 16 18 20 ________________ y 54 53 55 54 561 solutions Answer 375148 by Theo(3464)   on 2012-03-22 16:02:58 (Show Source): You can put this solution on YOUR website!you can try to find the line manually or you can use a calculator, like the TI-83 or TI-84 or even the TI-30X-IIS I used the TI-30X-IIS for this problem. The calculator told me that the equation for the best fit of this data is: y = .25x + 50.4 the correlation factor is .693375245 which is a pretty good positive fit. correlation factor of 1 would be a perfect positive fit. correlation factor of -1 would be a perfect negative fit. correlation factor of 0 would be the worst fit (neither negative or positive correlation can be derived). a table of your data and the derived data from the calculator generated best fit equation is shown below: ```x y calculator best fit y manual best fit y 12 54 53.4 14 53 53.9 16 55 54.4 18 54 54.9 20 56 55.4 ``` a picture of your data points and a straight line best fit that i did using the eyeball method gave a pretty close approximation of what the calculator did for me. you can also manually calculate the best fit linear equation using the method in the following link. http://www.statisticshowto.com/articles/how-to-find-a-linear-regression-equation/ that, however, is doing it the hard way and definitely not recommended unless you absolutely are required to do so. calculators are so much easier. it's a good learning tool, however, for you to appreciate how the calculator does it
 Linear-systems/590352: How do I solve 2x+3y=18 and x=2y-5 its one problem and I want to use the substitution method.thanks 1 solutions Answer 375120 by Theo(3464)   on 2012-03-22 14:50:21 (Show Source): You can put this solution on YOUR website!2x + 3y = 18 x = 2y-5 substitute 2y-5 for x in the first equation to get: 2(2y-5) + 3y = 18 simplify by removing parentheses to get: 4y - 10 + 3y = 18 combine like terms to get: 7y - 10 = 18 add 10 to both sides to get: 7y = 28 divide both sides by 7 to get: y = 4 substitute 4 for y in the second equation of x = 2y-5 to get: x = 2*4-5 which becomes: x = 8-5 which becomes: x = 3 substitute 3 for x and 4 for y in the first equation of 2x + 3y = 18 to get: 2*3 + 3*4 = 18 which becomes: 6 + 12 = 18 which becomes: 18 = 18 values for x and y are good. x = 3 y = 4
 Geometry_Word_Problems/590348: Hello, How do you set up an equation in algebra to show that the width of a rectangle is 3 ft shorter than the length and that the perimeter of the rectangle is 86 feet? 86=3w -6 I am soooo confused! Please help! bmccain@wowway.com1 solutions Answer 375114 by Theo(3464)   on 2012-03-22 14:35:33 (Show Source): You can put this solution on YOUR website!L = length of rectangle W = width of rectangle W = L - 3 P = perimeter = 86 feet. formula for perimeter: 2L + 2W = 86 substitute L-3 for W to get: 2L + 2(L-3) = 86 simplify by removing parentheses to get: 2L + 2L - 6 = 86 combine like terms to get: 4L - 6 = 86 add 6 to both sides to get: 4L = 92 divide both sides by 4 to get: L = 23 since W = L - 3, then: W = 20 formula for perimeter is: 2L + 2W = 86 substitute for L and W to get: 2*23 + 2*20 = 86 simplify to get: 46 + 40 = 86 combine like terms to get: 86 = 86 values for L and W are good.
 Trigonometry-basics/588572: I need to prove this problem: sec^2 t / tan t = (sec t)(csc t)1 solutions Answer 374542 by Theo(3464)   on 2012-03-19 00:42:28 (Show Source): You can put this solution on YOUR website!tan(t) = sin(t) / cos(t) sin(t) = 1/csc(t) cos(t) = 1/sec(t) by substitution, tan(t) = (1/csc(t)) / (1/sec(t)) this is equivalent to: tan(t) = (1/csc(t))*sec(t) which is equivalent to: tan(t) = sec(t)/csc(t) your original equation is: sec^2(t)/tan(t) = sec(t)*csc(t) multiply both sides of this equation by tan(t) to get: sec^2(t) = sec(t)*csc(t)*tan(t) substitute sec(t)/csc(t) for tan(t) to get sec^2(t) = sec(t)*csc(t)*sec(t)/csc(t) the csc(t) in the numerator and denominator cancel out and you are left with: sec^2(t) = sec(t)*sec(t) since sec(t)*sec(t) is equal to sec^2(t), you are done.
 Coordinate-system/588565: DTERMINE IF (1/3,-5)is a solution-3x-10y=511 solutions Answer 374541 by Theo(3464)   on 2012-03-19 00:20:26 (Show Source): You can put this solution on YOUR website!your equation is -3x - 10y = 51 x = (1/3) y = (-5) substitute in the equation to get: -3 * (1/3) - (10 * -5) = 51 simplify to get: -3/3 - (-50) = 51 simplify further to get: -1 + 50 = 51 combine like terms to get: 49 = 51 since this is not true, then those values are not solutions to the equation. if your equation was: 3x - 10y = 51, then: substituting would get you: 3*(1/3) - (10*(-5)) = 51 which would simplify to: 1 - (-50) = 51 which would simplify further to: 1 + 50 = 51 which would become: 51 = 51 in that case it would be a solution to the equation. it depends on what your equation is. the process is the same. the result is different if i assume your original equation is: -3x-10y=51 or whether i assume your original equation is: 3x-10y = 51 check your original equation to make sure what it is and you can find the answer depending on that. the process is the same regardless of the equation. you replace the variable x with its value and the replace the variable y with its value and then you solve. the coordinate pair of (1/3, -5) is the (x,y) coordinate pair which means that the value of x is equal to 1/3 and the value of y is -5.
 Miscellaneous_Word_Problems/588576: The resistance of a wire varies directly with its length and inversely with the square of its diameter. If 100 feet of wire with diameter 0.01 inch has a resistance of 90 ohms, what is the resistance of 68 feet of the same type of wire if its diameter is 0.02 inch?1 solutions Answer 374540 by Theo(3464)   on 2012-03-19 00:10:54 (Show Source): You can put this solution on YOUR website!let L equal the length of the wire and let D equal the diameter of the wire. let R equal the resistance of the wire. the formula would be: R = k * L / D^2 you know that 100 fee of wire with a diameter of .01 inches has a resistance of 90 ohms. you can use this fact to find k, since k is a constant and will not vary. your equation of: R = k * L / D^2 becomes: 90 = k * 100 / (.01)^2 if you multiply both sides of this equation by (.01)^2 / 100 then you get: 90 * (.01)^2 / 100 = k solve for k to get: k = .00009 your formula of: R = k * L / D^2 becomes: R = .00009 * L / D^2 test this formula out with your original measurements to get: R = .00009 * 100 / (.01)^2 and you will get: R = 90 ohms. use this formula with your new measurements and you will get: R = .00009 * 68 / (.02)^2 which becomes: R = 15.3 ohms.