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 Geometry_Word_Problems/255274: The surface area of a sphere is 3600 pi square inches. If v represents the volume of the sphere in cubic inches, what is v/pi ?1 solutions Answer 187543 by Theo(3464)   on 2010-01-10 08:45:59 (Show Source): You can put this solution on YOUR website!formula for the surface area of a sphere is: s = 4 * pi * r^2 formula for the volume of a sphere is: v = (4/3) * pi * r^3 since s = 4 * pi * r^2, and since s = 3600 * pi, then we get: 3600 * pi = 4 * pi * r^2 divide both sides of this equation by pi to get: 3600 = 4 * r^2 divide both sides of this equation by 4 to get: 900 = r^2 take square root of both sides of this equation to get: r = 30 We want to find what v/pi equals. since v = (4/3) * pi * r^3, and since r = 30, we get: v = (4/3) * pi * (30)^3 which becomes: v = (4/3) * pi * 27000 divide both sides of this equation by pi to get: v/pi = (4/3) * 27000 = 36000. our answer is: v/pi = 36000
 Percentage-and-ratio-word-problems/255188: If you mix fruit juice and bubbly water in a ratio of 3 to 5 to make a punch then how many liters of juice and how many liters of bubbly water will you need to make 24 liters. Please explain with a strip diagram.1 solutions Answer 187540 by Theo(3464)   on 2010-01-10 08:31:00 (Show Source): You can put this solution on YOUR website!f = number of liters of fruit juice. b = nume of liters of bubbly water. ratio f/b = 3/5 total liters = 24. your equation is f + b = 24. since f/b = 3/5, thie means that: f = 3b/5 equation of f + b = 24 becomes: 3b/5 + b = 24 multiply both sides of this equation by 5 to get: 3b + 5b = 120 which becomes: 8b = 120 which becomes: b = 120/8 = 15. since f + b = 24, this means that f = 24-15 = 9. you have f = 9 and b = 15 f + b = 9 + 15 = 24 f/b = 9/15 = 3/5 ratio holds and the sum is correct. if you used a strip diagram, you might do the following: let 1 strip represent 3 liters of fruit juice. this strip would be divided into 3 equal sections. let another strip represent 5 liters of bubbly water. this strip would be divided into 5 equal sections. each section would represent 1 liter. first strip would look like fff second strip would look like bbbbb since the ratio has to remain constant, then combine these strips into one long strip that is composed of 3 sections of f and 5 sections of b. that would look like: fffbbbbb = 8 liters. since you want 24 liters, how many times does this strip divide into 24. answer is 3. you get: fffbbbbb + fffbbbbb + fffbbbbb = 24 liters. add up the f's and you get 9. add up the b's and you get 15. you have 9 liters of fruite juice and 15 liters of bubbly water to make a total of 24 liters of mixture.
 Finance/255266: I'm stuck on the following problem: What annual interest rate compounded annually is required to provide a 21% increase in the initial investment after 2 years. I tried to use the formula S = P(1 + i) to the n power. I put 10,000 in for P as the original investment (just made it up), then I put 12,100 as S the total investment showing an increase of 21%. Then I filled in 4 for n because the interest is compounded annually! Then try and solve for i Not sure if I'm on the right track!!1 solutions Answer 187534 by Theo(3464)   on 2010-01-10 07:48:50 (Show Source): You can put this solution on YOUR website!looks like you started off ok. your formula is: S = P * (1+i)^n you used 10,000 for P and 1.21 * 10,000 = 12100 for S. that's correct but not necessary because you could just as easily have used 1 for P and 1.21 for S. I don't know why you filled in 4 for n. if compounding is annually, then n should have been 2 to represent 2 years. using your numbers for S and P, your formula should have become: 12100 = 10000 * (1+x)^2 x is the annual interest rate. divide both sides of this equation by 10000 to get: 1.21 = (1+x)^2 take the square root of both sides of this equation to get: subtract 1 from both sidesof this equation to get: solve for x to get: x = .1 looks like you are looking for 10% a year growth compounded annually for 2 years. to confirm, substitute in your original equation to get: 12100 = 10000 * (1+x)^2 becomes: 12100 = 10000 * (1.1)^2 simplify to get: 12100 = 12100 confirming the answer is good.
 Geometric_formulas/255178: My homework is on geometric probability. One of my problems has to do with a dart board. I am given that the Diameter of the outer double ring = 41 cm. the diameter of the inner double ring = 36 cm. the diameter of the outer triple ring = 25.6 cm. the diameter of the inner triple ring = 21.2 cm. the diamer of the outer bulls eye = 4.4 cm. and finally the diameter of the inner bull's eye = 1.4 cm. My question asks What is A) what is the probability that the dart will land IN the double ring B) What is the probability of hitting the single bull's eye and finally C) what is the probability of hitting the triple 20 ring. Please help as much as possible, I'd really appreciate it. Thank you! ** Note: The numbers go from 1- 20 on a dart board if that helps at all.1 solutions Answer 187533 by Theo(3464)   on 2010-01-10 07:34:35 (Show Source): You can put this solution on YOUR website!if we assume that you can hit anywhere in the scoring area of the dartboard 100% of the time, then the total scoring area of the dartboard represents 100% probability. any area less than that on the dartboard means the probability of hitting that area = that area divided by the total area of the dartboard. total area of the dartboard is equal to the area of the circle from the center of the circle to the outer double ring line. the area of a circle is equal to pi*r^2 where r is the radius. singe the diameter of a circle is equal to 2 times the radius of a circle, then the area of a circle can be shown as pi*(d/2)^2 where d is the diameter. the scoring area of the dartboard is equal to pi * (41/2)^2 = pi * 20.5^2 = 1320.254313 cm^2. if you look at the picture in the attached link, that's the area from the center of the dartboard to the double ring line. that's the total scoring area of the dartboard and we are assuming you will hit in the scoring area of the dartboard 100% of the time. ------------------------------------------------------------------- if you want to hit in the double scoring area, that's the area between the outer double ring line and the inner double ring line. the diameter of the outer double ring line is 41 cm. the diameter of the innter duble ring line is 36 cm. the double ring area is the area from the center of the circle to the outer double ring line minus the area from the center of the circle to the inner double ring line. the area between those lines is equal to pi*(41/2)^2 - pi*(36/2)^2 = pi*20.5^2 - pi*18^2 = 1320.254313 - 1017.87602 = 302.3782929 cm^2. the probability of hitting in the double scoring area is therefore equal to: 302.3782929 cm^2. / 1320.254313 cm^2 = .229030339 ----------------------------------------------------------------------- the probability of hitting the single bullseye depends on what you call the single bullseye. you have an outer bullseye line and an inner bullseye line. the very center is the double bullseye which means the single bullseye area must be between the outer bullseye line and the inner bullseye line. this equals the area from the center of the circle to the outer bullseye line minus the area from the center of the circle to the inner bullseye line. the area of the circle from the center to the outer bullseye line is equal to: pi*(4.4/2)^2 = pi*2.2^2 = 15.20530844 cm^2. the area of the circle from the center to the inner bullseye line is equal to: pi*(1.4/2)^2 = pi*.7^2 = 1.5393804 cm^2. the area between the outer bullseye line and the inner bullseye line is equal to 15.20530844 cm^2 - 1.5393804 cm^2 = 13.66592804 cm^2 the probability of hitting in the single bullseye area is therefore equal to: 13.66592804 cm^2 / 1320.254313 = .010350982 ----------------------------------------------------------------- the area of the triple 20 is equal to 1/20th of the triple area. the triple area is the area between the outer triple line and the inner triple line. the area from the center of the circle to the outer triple line is equal to: pi * (25.6/2)^2 = pi * 12.8^2 = 514.7185404 the area from the center of the circle to the inner triple line is equal to: pi * (21.2/2)^2 = pi * 10.6^2 = 352.9893506 the triple area is the area between these 2 lines which equals 514.7185404 minus 352.9893506 = 161.7291898 cm^2 the triple 20 area would be 1/20th of this which would equal 8.086459493 the probability of hitting into the triple 20 area is therefore equal to: 8.086459493 / 1320.254313 = .006124926 the whole thing depends on the folloowing: *** you will hit the scoring area of the dartboard 100% of the time. *** the probability of hitting one area of the scoring area of the dartboard is the same as the probability of hitting any other area of the scoring area of the dartboard. *** the probability of hitting any area within the scoring area of the dartboard is based on the size of that area divided by the size of the scoring area of the dartboard. any change in these assumptions changes the outcome.
 Miscellaneous_Word_Problems/255164: In 1995, Sweden and Norway had more women members of parliament (MPs) than any other democratic countries. Combined, the two countries had 206 women in parliaments. Taking double the number of women MPs in Sweden and adding this to triple the number of women MPs in Norway results in 477. a. How many women MPs are there in each country? b. The total MPs in Sweden is 349 and in Norway the number is 165. What percent (to the nearest whole percent) of the total MPs are women for each of the two countries?1 solutions Answer 187531 by Theo(3464)   on 2010-01-10 05:36:06 (Show Source): You can put this solution on YOUR website!let s = number of women members of parliament in sweden. let n = number of women members of parliament in norway. s + n = 206 2s + 3n = 477 solve for s in the first equation to get s = 206 - n substitute in the second equation to get 2*(206-n) + 3n = 477 simplify to get 412 - 2n + 3n = 477 combine like terms to get 412 + n = 477 subtract 412 from both sides of equation to get n = 477 - 412 = 65 if n = 65, then s = 206 - 65 = 141 equation of s + n = 206 becomes 141 + 65 = 206 which is true. equation of 2s + 3n = 477 becomes 2*141 + 3*65 = 477 becomes 282 = 195 = 477 which is also true. numbers look good. s = 141 = number of women members of parliament in sweden. n = 65 = number of women members of parliament in norway. total members of parliament in sweden is 349. total women members of parliament in sweden is 141. 141/349 = .404011461 = 40% to the nearest whole percent. total members of parliament in norway is 165. total women members of parliament in norway is 65. 65/165 = .393939394 = 39% to the nearest whole percent.
 Miscellaneous_Word_Problems/255254: What is the smallest positive integer n so that 13n leaves a remainder of 1 when divided by 11?1 solutions Answer 187528 by Theo(3464)   on 2010-01-10 05:18:19 (Show Source): You can put this solution on YOUR website!13/11 = 1+2 26/11 = 2+4 39/11 = 3+6 52/11 = 4+8 65/11 = 5+10 78/11 = 7+1 *****************************8 looks like the number is 6 6*13 = 78 78/11 = 7 + a remainder of 1. 7*11 = 77 + 1 = 78
 Numbers_Word_Problems/255255: What is the one’s digit of the number 7^2005 ?1 solutions Answer 187527 by Theo(3464)   on 2010-01-10 05:02:44 (Show Source): You can put this solution on YOUR website!7^1 = 7 7^2 = 49 7^3 = 343 7^4 = 2401 7^5 = 16807 7^6 = 117649 7^7 = 823543 7^8 = 5764801 7^9 = 40353607 7^10 = 282475249 7^11 = 1977326743 the last digit is repeating in a pattern of: 7,9,3,1, 7,9,3,1, 7,9,3,1, to find out what the last digit is, you need to make a formula. the formula would be taking the larger exponent and breaking it down into the smaller exponent example: suppose your number is 7^6. 6/4 = 2 and 7^2 = 49 so the last digit is 9. look at 7^6 above and you can see that the last digit is 9. suppose your number is 7^11. 11/4 = 2 with a remainder of 3. 7^3 = 343 so the last digit is 3. look up 7^11 above and you can see that the last digit is 3. you can use this formula for larger numbers even though you can't check them because the calculator doesn't carry enough digits. I used excel and the highest I could go was 7^17 as shown below: ```base exponent number last digit 7 1 7 7 7 2 49 9 7 3 343 3 7 4 2401 1 7 5 16807 7 7 6 117649 9 7 7 823543 3 7 8 5764801 1 7 9 40353607 7 7 10 282475249 9 7 11 1977326743 3 7 12 13841287201 1 7 13 96889010407 7 7 14 678223072849 9 7 15 4747561509943 3 7 16 33232930569601 1 7 17 232630513987207 7 7 18 9 7 19 3 7 20 1 7 21 etc..... ``` for a final check, try 7^16. 16/4 = 4 with a remainder of 0. 7^0 = 1 so the remainder should be 1. it is, as shown in the table from excel above. note that 16 / 4 = 4 with a remainder of 0. it also equals 3 with a remainder of 4 which confirms that the last digit is 1. In your problem, the number is 7^2005 divide 2005 by 4 to get 501 with a remainder of 1. 7^1 = 7, so the last digit of 7^2005 = 1
 Inequalities/255240: i need help with setting up this problem because im not sure how: a home owner is selling her house. she must pay 7% of the selling price to her real estate agent after the house is sold define a variable and write and solve an inequality to find what the selling price of her house must be to make at least 140,000\$ round to nearest dollar. -thank you 1 solutions Answer 187523 by Theo(3464)   on 2010-01-10 04:31:48 (Show Source): You can put this solution on YOUR website!let x = selling price of house. she must pay .07*x to the real estate agent. she wants to make \$140,000 after she had paid the real estate agent. equation is: x - .07*x = 140000 this becomes x * (1-.07) = 140000 this becomes x * .93 = 140000 divide both sides of this equation by .93 to get: x = 140000 / .93 = 150537.6344 if she sells the house for that price, then the broker gets: .07*150537.6344 = 10537.63441 selling price is 150537.6344 broker's commission is 10537.63441 150537.6344 - 10537.63441 = 140000. your answer is: she must sell the house for \$150,537 in order to make \$140,000, assuming the real estate agent gets 7% of the selling price.
 Graphs/255227: 7r-6s=-22 6r+7s=54 solve. i think i need coordinates so i can graph it. if that makes any sense. 1 solutions Answer 187520 by Theo(3464)   on 2010-01-10 04:10:23 (Show Source): You can put this solution on YOUR website!your equations are: 7r - 6s = -22 6r + 7s = 54 if you let: y = s and x = r, your equations become: 7x - 6y = -22 6x + 7y = 54 now you can graph them on an xy plane where x is the horizontal axis and y is the vertical axis. in order to graph them, you have to solve for y in both equations. first equation becomes y = (7/6)x + (22/6) second equation becomes y = -(6/7)x + (54/7) graph of these equations looks like this: to graph them manually, you would need coordinates, but you would only need 2 for each equation since they are straight lines. if you let x = 0, one of the coordinates pops up easily. then just pick any other value for x, like 5 or 10, and you get your second coordinate. then just draw a straight line passing through those 2 points. these equations are perpendicular to each other. after you solved for y, they are in slope intercept form. slope intercept form is y = mx + b where m is the slope and b is the y-intercept. y-intercept is the value of y when x = 0. slope for the first equation is (7/6). slope for he second equation is -(6/7). negative reciprocal of (7/6) is -(6/7) telling you that these lines are [perpendicular to each other. the form that they started off in is the standard form of the equation of a straight line which is ax + by = c if you graph them manually, you could keep them as r and s. just make r the horizontal axis and s the vertical axis. I had to convert them to x and y because the graphing function required them to be x and y. keeping them as r and s, then the slope intercept form of the equations is: s = (7/6)r + (22/6) s = -(6/7)r + (54/7)
 Quadratic_Equations/255064: The sides of an equilateral triangle are shortened by 12 units,13 units and 14 units respectively and a right angled triangle is formed.Find the side of the equilateral triangle.1 solutions Answer 187365 by Theo(3464)   on 2010-01-09 08:51:07 (Show Source): You can put this solution on YOUR website!x = 17. 17 - 14 = 3 17 - 13 = 4 17 - 12 = 5 345 triangle is a right triangle. solve by use of pythatogrean formula. a^2 + b^2 = c^2 x is the side of the equilateral triangle. (x-12) = largest side which would be equal to c. (x-13) = midsize side which would be equal to b. (x-14) = smallest side which would be equal to a. a^2 + b^2 = c^2 becomes: (x-14)^2 + (x-13)^2 = (x-12)^2 this becomes x^2 - 28x + 196 + x^2 - 26x + 169 = x^2 - 24x + 144 combine like terms to get: 2x^2 - 54x + 365 = x^2 - 24x + 144 subtract x^2 from both sides of equation to get: x^2 - 54x + 365 = -24x + 144 add 24x to both sides of equation to get: x^2 - 30x + 365 = 144 subtract 144 from both sides of equation to get: x^2 - 30x + 221 = 0 factor this quadratic equation to get: (x-17) * (x-13) = 0 this makes x = 17 or x = 13. x can't be 13 because then one of the sides of the triangle will be 0. x has to be 17. when x is 17, you get: x-12 = 17-12 = 5 x-13 = 17-13 = 4 x-14 = 17-14 = 3 that's your 345 triangle which is a right triangle. side of the equilateral triangle is 17.
 Miscellaneous_Word_Problems/255053: Lonnie has a large supply of quarters , dimes, nickels, and pennies. In how many ways could she make change for 50 cents? 1 solutions Answer 187348 by Theo(3464)   on 2010-01-09 07:00:05 (Show Source): You can put this solution on YOUR website!1 quarter = 25 cents 1 dime = 10 cents 1 nickel = 5 cents 1 penny = 1 cent it would make sense to make a table of this working our way down from the highest denomination to the lowest denomination. the total value of each row is equal to 50 cents. ``` quarters dimes nickels pennies 2 0 0 0 1 2 1 0 1 2 0 5 1 1 3 0 1 1 2 5 1 1 1 10 *** example 1 1 1 0 15 1 0 5 0 1 0 4 5 1 0 3 10 1 0 2 15 1 0 1 20 1 0 0 25 0 5 0 0 0 4 2 0 0 4 1 5 0 4 0 10 0 3 4 0 0 3 3 5 *** example 2 0 3 2 10 0 3 1 15 0 3 0 20 0 2 6 0 0 2 5 5 0 2 4 10 0 2 3 15 0 2 2 20 0 2 1 25 0 2 0 30 0 1 8 0 0 1 7 5 0 1 6 10 0 1 5 15 0 1 4 20 0 1 3 25 0 1 2 30 0 1 1 35 0 1 0 40 0 0 10 0 0 0 9 5 0 0 8 10 0 0 7 15 0 0 6 20 0 0 5 25 *** example 3 0 0 4 30 0 0 3 35 0 0 2 40 0 0 1 45 0 0 0 50 ``` this counts out to be 49 possible combinations. I couldn't figure out a formula to use so I just tabulated all the possible combinations I could think of. The total is 49. I believe I have them all covered. Any combination you can think of should be in the table. Example 1: 1 quarter 1 dime 1 nickel 10 pennies. start with the quarter column and scan down until you see 1. shift to the dime column and scan further down until you find 1. shift to the nickel column and scan further down until you find 1. shift to the penny column and you should see 10. Example 2: 3 dimes 3 nickels 5 pennies. start with the quarter column and scan down until you see 0. shift to the dime column and scan further down until you find 3. shift to the nickel column and scan further down until you find 3. shift to the penny column ahd you should see 5. Example 3: 5 nickels 25 pennies. start with the quarter column and scan down until you see 0. shift to the dime column and scan further down until you find 0. shift to the nickel column and scan further down until you find 5. shift to the penny column and you should see 25.
 Numbers_Word_Problems/255054: How many different four-digit numbers can be formed using the digits 1, 1, 9, and 9?1 solutions Answer 187325 by Theo(3464)   on 2010-01-09 05:23:44 (Show Source): You can put this solution on YOUR website!should be equal to (4!)/(2!*2!) = (4*3*2*1)/(2*1*2*1) = 6 let's see what they would be: 1199 9911 1919 9191 1991 9119 this winds up being a permutation with some of the elements being the same. formula for a permutation is n! if x of the n are the same, then the formula becomes n!/x! if, in addition to x being the same, y are also the same, then the formula becomes n!/(x!*y!) in your problem, n was equal to 4. x was equal to 2 because you had 2 ones. y was also equal to 2 because you had 2 nines. n!/(x!*y^1) became 4!/(2!*2!) to see what the difference is, look at 3 letters (a,b,c) how many ways can they be formed? 3! = 3*2*1 = 6 ways because each of the letters is different. they are: abc acb bac bca cab cba take the same 3 letters and make 2 of them the same. let's assume a and b become the same letter d. you would then have ddc how many ways can they be formed. we have n = 3 and x = 2 and our formula is n!/x! = 3!/2! = 3 the number of ways they can be formed is 3 as follows: ddc dcd cdd please keep in mind that we are talking about permutations here, and not combinations. with permutations, order is important. the same elements in a different order are a different set. with combinations, order is not important. each set has to have different elements in it. take the set of abc. with combinations, this forms one set. the same 3 elements can only be used once regardless of what order they are in the set. with permutations, this forms 6 sets. the same 3 elements can be used 6 times in a different order each time. the number of combinations possible would be 1. the numer of permutations possible would be 6.
 Rational-functions/254942: The body mass index, I, can be used to determine an individual’s risk for heart disease. An index less than 25 indicates a low risk. The body mass index is given by the formula, or model. I=700W/H2, where W=weight, in pounds, and H=height, in inches. Jerome is 67 inches tall. What weights will keep his body mass index between 23 and 33? 1 solutions Answer 187322 by Theo(3464)   on 2010-01-09 05:09:36 (Show Source): You can put this solution on YOUR website!I = 700*W/H2 W = weight in pounds H = height in inches < 25 = low risk I presume when you says H2, you mean . Jerome is 67 inches tall. the formula to work with is: substitute 67 for h to get: In order for his index to be between 23 and 33, this formula needs to be modified as follows: and for greater than or equal to 23, the formula is: multiply both sides of this equation by (67)^2 = 4489 to get: divide both sides of this equation by 700 to get Solve for W to get: W >= 147.4957143 Similarly, the second equation of: becomes: Solve for W to get: W <= 211.6242857 To the nearest tenth of a pound, Jerome needs to keep his weight between 147.5 and 211.6 pounds in order to keep his index between 23 and 33.
 Geometry_proofs/254834: how would you design a circuit for ahall light that has switches at both ends of hall such that the light can be turned on or off from either switch. I have to use this problem for this question Electrical circuits are designed using truth tables.circuit consists of switches. And wired in series as p^q. two switches wired in parallel can be use as p v q. Can you show me how to do this problem using the truth table1 solutions Answer 187167 by Theo(3464)   on 2010-01-08 10:41:28 (Show Source): You can put this solution on YOUR website!a diagram of your switch could look like this: ``` A1 A2 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx battery xxxxxxxxxxxxxxx C1 C2 xxxx light xxxxx ground xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx B1 B2 ``` you have battery coming in from the left going through the switches and then through the light and then to ground. the following connections are possible through switch 1 and switch 2. ``` switch 1 switch 2 light C1 to A1 C2 to A2 ON C1 to A1 C2 to B2 OFF C1 to B1 C2 to B2 ON C1 to B1 C2 to A2 OFF ``` if we let: C1 to A1 = UP C1 to B1 = DOWN C2 to A2 = UP C2 to B2 = DOWN then the table becomes: ``` switch 1 switch 2 light UP UP ON UP DOWN OFF DOWN DOWN ON DOWN UP OFF ``` this becomes the basis for your truth table. in order to construct the truth tables, we will have 4 variables. they will be U1, U2, D1, D2 U1 means switch 1 is up. U2 means switch 2 is up. U3 means switch 3 is down. U4 means switch 4 is down. T means True means the switch is as indicated. F means False means the switch is not as indicated. the 5th column will be L representing the light. T means True means the light is on. F means False means the light is off. Here's what your truth table would look like if constructed this way. ``` U1 D1 U2 D2 L T F T F T T F F T F F T F T T F T T F F ``` if the implication is that when a switch is not in the up position, then it has to be in the down position, we can simplify the truth table to be as follows: ``` U1 U2 L T T T T F F F F T F T F ``` U1 being true means that switch 1 is in the up position. U1 being false means that switch 1 is in the down position. Similarly for U2 the logic statements for this truth table would look like: U1 & U2 => L this means that the state of U1 and the state of U2 both define the state of L. Hopefully this is what you are looking for, or at least something similar to it.
 Permutations/254821: how many possible 5-card hands can make out of 2 of 1 kinf, 2 of another kind, and 1 0ther card in a regular 52-card deck??1 solutions Answer 187133 by Theo(3464)   on 2010-01-08 08:39:43 (Show Source): You can put this solution on YOUR website!you want 2 of one kind, 2 of another kind, and any card for the 5th. your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs. there are 4 suits in a deck and each suit has 13 cards making a total of 4 * 13 = 52. That means you have 4 aces, 4 kings, 4 queens, 4 jacks, 4 tens, etc. down to 4 deuces. since there are 4 suits of the same card, and you need to get 2 of them, then your possible combinations are C(4,2) = 4!/2!2! = 6 possible ways for each pair. assume it's a pair of aces. you can get them as follows: ace of clubs and ace of spades ace of clubs and ace of diamonds ace of clubs and ace of hearts = 3 + ace of spades and ace of diamonds ace of spades and ace of hearts = 5 + ace of diamonds and ace of hearts = 6 so for each pair, you can do it in 6 ways. since there are 13 sets of the same card, the total number of combinations of two different pairs you can make from this deck of card is C(13,2) = 13!/(11!*2!) = 13*12/2 = 13*6 = 78. some examples would be: kings and queens kings and jacks kings and tens etc. once you have your 2 pairs, the total number of ways you can draw the 5th card is (52-8) * 1 = 44 total possible combinations of different pairings would therefore be: 6*6*78*44 = 123552 that's 78 combinations of different pairs. each pair can be derived in 6 ways. each combination of pairs can therefore be derived in 6*6 = 36 ways. you get 6*6*78 possible combinations of pairs. for each of those possible combinations of pairs, there are 44 possible ways that the fifth card can happen. ------------------------------------------------------------------------- to make this easier to understand in an example, let's assume your deck of cards had only kings, queens, and jacks in it. let's also assume there are only 3 suits (clubs, spades, hearts). that would be a total of 9 cards in the deck with 3 suits. you want 2 of one kind, 2 of another kind, and any card for the 5th. your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs. there are 3 suits in this deck and each suit has 3 cards making a total of 3 * 3 = 9 cards in the deck. That means you have 3 kings 3 queens and 3 jacks. since there are 3 of each kind, and you need to get 2 of them to make a pair, then your possible combinations for each pair are C(3,2) = 3!/1!2! = 3. assume it's a pair of kings you can get them as follows: king of clubs and king of spades king of clubs and king of hearts king of spades and king of hearts so for each pair, you can do it in 3 ways. the total number of combinations of two different pairing you can make from this deck of card is C(3,2) = 3!/(1!*2!) = 3*2*1/1*2*1 = 6/2 = 3. those possible combinations of pairings would be: kings and queens kings and jacks queens and jacks once you have your 2 pairs, the total number of ways you can draw the 5th card is (9-6) * 1 = 3 for kings and queens, the fifth card can be: jack of clubs jack of spades jack of hearts total possible number of combinations of pairing would therefore be: 3*3*3*3 = 81 you have 3 possible combinations of two pairs each. each pair has 3 possible ways it can be achieved. each combination of pairs can therefore happen in 3*3 = 9 ways. total possible combinations of pairing is therefore 9*3 = 27 possible ways. for each of these possible combinations of pairings, the fifth card can be picked in 3 ways. that's a total of 27*3 = 81 possible ways. here are the posible combinations for just 1 of the 3 possible combinations of pairings. the 3 possible combination of pairings are: kings and queens kings and jacks queens and jacks this pairing will be for kings and queens. let k = king let q = queen let j = jack let 1 = clubs let 2 = spades let 3 = hearts you get: k1 = king of clubs, k2 = king of spades, k3 = king of hearts q1 = queen of clubs, q2 = queen of spades, q3 = queen of hearts j1 = jack of clubs, j2 = jack of spades, j3 = jack of hearts for kings paired with queens, possible combinations of pairings are: ``` k1 k2 q1 q2 j1 k1 k2 q1 q3 j1 k1 k2 q2 q3 j1 k1 k2 q1 q2 j2 k1 k2 q1 q3 j2 k1 k2 q2 q3 j2 k1 k2 q1 q2 j3 k1 k2 q1 q3 j3 k1 k2 q2 q3 j3 k1 k3 q1 q2 j1 k1 k3 q1 q3 j1 k1 k3 q2 q3 j1 k1 k3 q1 q2 j2 k1 k3 q1 q3 j2 k1 k3 q2 q3 j2 k1 k3 q1 q2 j3 k1 k3 q1 q3 j3 k1 k3 q2 q3 j3 k2 k3 q1 q2 j1 k2 k3 q1 q3 j1 k2 k3 q2 q3 j1 k2 k3 q1 q2 j2 k2 k3 q1 q3 j2 k2 k3 q2 q3 j2 k2 k3 q1 q2 j3 k2 k3 q1 q3 j3 k2 k3 q2 q3 j3 ``` for k1k2 you have 3 possible combinations with queens. those would be q1q2, q1q3, q2q3. for kik3 you have 3 more possible combinations with queens. those would be q1q2, q1q3, q2q3. for k2k3 you have 3 more possible combinations with queens. those would be q1q2, q1q3, q2q3. that's a total of 3*3 = 9 possible combinations with kings and queens. for each one of those combinations you have 3 possible combinations of the fifth card. it can be j1, j2, or j3. that makes a total of 9*3 = 27 possible combinations for kings and queens alone. you can see that for kings and queens there are 3*3*3 = 27 possible hands that can be dealt. kings can also be paired with jacks which would yield another 27. queens can also be paired with jacks which would yield another 27. total possible pairings would be 81. same principle applies to the larger deck of 4 suits with 13 cards in each suit. with the larger deck (52 cards, 13 suits, 4 of the same card in each suit), you get the following: you have a total of 6 possible ways to get each pairing. each combination of pairing can happen in 6*6 = 36 ways. there are a total of 78 possible combinations of pairings. that makes a total of 78*36 = 2808 possible combinations of pairings. for each of those possible combinations of pairings, you have a choice of 44 different cards to be the fifth card. that makes a total of 2808 * 44 = 123552 possible ways you can get a hand with 2 pairs only.
 Geometry_Word_Problems/254435: 1.)A 12-foot ladder is leaning against the wall. When a painter P was 2/3 of the way of the ladder he dropped his brush which fell directly below him, 1 ft away from the wall. How far was the base of the ladder on the wall? 2.)You are flying a kite and have let out 1000 feet of string. The kite string makes a 42º angle with the ground. How high is the kite?1 solutions Answer 186760 by Theo(3464)   on 2010-01-07 07:44:32 (Show Source): You can put this solution on YOUR website!12 foot ladder leaning against a wall. painter is 2/3 of the way up when he drops his brush. brush lands directly beneath him 1 foot away from the wall. how far from the base of the ladder is the wall. 2/3 of the way up is 8 feet on the ladder. diagram of the ladder against the wall is shown below. ``` A x x x x x x x 4 feet x x x x x x F x x x x x x x x D x x x x x x x x x x x x x x x x x x x x x 8 feet x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x C E B ``` A is the point where the ladder touches the wall. B is the point where the ladder touches the ground (base of the ladder). C is the point where the wall touches the bround (base of the wall). D is where the painter is. E is where the brush drops F is where a line from D parallel to the ground touches the wall. you have formed 2 similar triangles. they are triangle ABC and triangle ADF the distance from C to E on the ground is 1 foot. the distance from F to D parallel to the ground is also 1 foot. since the triangle are similar, then their corresponding sides are proportional. this means that AD / FD = AB / CB since AD is 4 feet and AB is 12 feet and FD is 1 foot, this formula becomes: 4 / 1 = 12 / CB cross multiply to get: 4*CB = 1*12 divide both sides of this equation by 4 to get: CB = 12/4 = 3 your answer is that the distance from the base of the ladder to the base of the wall is 3 feet. -------------------------------------------------------------------- kite is 1000 feet long from the flyer of the kite to the kite along the length of the string. angle kite makes to the ground is 42 degrees. this forms a triangle as shown below: ``` A x x x x x x x x x x x x x x x x x 1000 feet x x x x x x x x x x x x x x x x x 42 degrees x x x x x x x x x x x x x x x x x x x x x x x C B ``` point A is the kite. point B is the person flying the kite. point C is a point directly beneath the kite perpendicular to the ground. the ground is represented by line segment CB length of line segment AB is 1000 feet. the height of the kite is represwented by line segment AC. tangent of 42 degrees = opposite / hypotenuse = AC / AB = AC / 1000 tan(42) = AC / 1000 multiply both sides of this equation by 1000 to get: 1000 * tan(42) = AC = 900.4040443. height of the kite is 900.4040443 feet.