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# Recent problems solved by 'Theo'

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 Linear_Equations_And_Systems_Word_Problems/302169: in 1992 life expectancy of males was 69.1 years in 1996 it was 72.9 years let e represent the life expectancy in year t and let t represent the # of years since 1992. I do not need the problem solved only the linear function that fits the data given (Ihave trouble figuring out the formulas of word problems but once I have that I do well at figuring them out). I have so far: E(t) = ?t + ? 1 solutions Answer 216699 by Theo(3464)   on 2010-05-09 06:27:22 (Show Source): You can put this solution on YOUR website!If you make 1992 = 0, the 1996 should equal to 4 because 1996 - 1992 = 4 and 4 - 0 = 4. You have 2 points of data. They are: (0,69.1) (4,72.9) If you let x = number of years and y = age expectancy, then you have an equation that looks like: y = m*x + b The slope of this equation is m which would be the change in life expectancy divided by the change in years. Using your 2 points of data, this would become: (72.9 - 69.1) / (4 - 0) = 3.8 / 4 = .95 This means that the life expectancy increases by .95 years every year. 4 * .95 = 3.8. You start with 69.1 in year 0 and 4 years later you have 69.1 + 3.8 = 72.9 Now that you have the slope, you can fill that in the equation to get: y = .95*x + b b is the y-intercept. You already know that b will be 69.1, but you can also solve for it to confirm. You can pick either of the 2 points to replace x and y with in the equation to solve for b. pick the point (0,69.1). y = .95*x + b becomes 69.1 = .95*0 + b which becomes b = 69.1. Your equation becomes y = .95*x + 69.1 Now you can set y equal to e(x) which makes your equation equal to e(x) = .95*x + 69.1 If you want to change x to t, then your equation becomes e(t) = .95*t + 69.1 Either way, that's your equation. The key to solving this problem is to determine from the data that is given, the change in life expectancy per year. That was done when we solved for the slope of the equation of the straight line above. The slope was the change in life expectancy divided by the change in years which resulted in the change in life expectancy per year.
 Probability-and-statistics/302159: 3. Which of the following experiments does NOT have equally likely outcomes? Choose a number at random from 1 to 7. Toss a coin. Choose a letter at random from the word SCHOOL. None of the above. RESULTS BOX: I am a substitute teacher who teaches K-6. I can complete all the probability problems except this one that I found on a web-site. I must have a mental block.1 solutions Answer 216681 by Theo(3464)   on 2010-05-09 04:18:25 (Show Source): You can put this solution on YOUR website!Choose a number at random from 1 to 7. Toss a coin. Choose a letter at random from the word SCHOOL. None of the above. I believe the third choice would not have an equally likely probability for each of the letters to be picked. For the first choice, the numbers are 1,2,3,4,5,6,7 with equal probability to pick any one of them because there is only one of each number. There are 7 letters and each has a probability of 1/7 to be picked. For the second choice, the possibilities are heads or tail with equal probability to pick any one of them because there is only one of each possibility. There are 2 possible outcomes and each has a probability of 1/2 to be picked. With the word SCHOOL, the probability for the letter O to come up is two times the probability for any of the other letters which have only one of each. Total letters are 6. Probability of S = 1/6 Probability of C = 1/6 Probability of H = 1/6 Probability of O = 2/6 ***** Probability of L = 1/6
 Graphs/301230: How do you solve a question that says find the rate of change for each linear function? Eg: The cost of cheese varies directly with the number of pounds bough. If 2 pounds cost \$8.40, find the cost of 3.5 pounds. 1 solutions Answer 215982 by Theo(3464)   on 2010-05-06 06:14:25 (Show Source): You can put this solution on YOUR website!Use the direct variation formula of y = k*x Solve for k first using the known values of x and y. Let x = 2 and y = 8.4 Formula becomes: 8.4 = k*2 Divide both sides of this equation by 2 to get: k = 4.2 k winds up being the slope of your straight line equation. The formula for a straight line equation is y = m*x + b m is the slope and b is the y-intercept. The slope is 4.2. The y-intercept is found by substituting known values for x and y in the equation and solving for b. the known values are (x,y) = (2,8.4) Equation of y = 4.2*x + b becomes 8.4 = 4.2*2 + b which becomes 8.4 = 8.4 + b. Solve for b to get b = 0. Your straight line equation is y = 4.2*x Graph this equation to get: Horizontal line at y = 8.4 intersects the graph of the equation of the line at x = 2. Horizontal line at y = 14.7 intersects the graph of the equation of the lin at x = 3.5. This confirms the equation is good. When x = 2, y = 8.4 When x = 3.5, y = 14.7 The rate of change is the slope of the line which is 4.2. This means that for every change of 1 in the value of x, y changes by 4.2.
 Quadratic-relations-and-conic-sections/298194: Write the equation. Hyperbola with C(-2,0) V(-2,3) y= +/- .5x I am pretty sure C stands for center and V for vertex. There are two vertices so the second one should be at (-2,-3) I think. I am not sure what to do with the y = +/- .5x though. 1 solutions Answer 214623 by Theo(3464)   on 2010-04-29 10:24:39 (Show Source): You can put this solution on YOUR website!The general equation for a horizontally oriented hyperbola would be: The center of your hyperbola is at C = (-2,0). The vertices of this hyperbola are probably at V1 = (-2,-3) and V2 = (-2,3) This, I believe, makes this a vertically oriented hyperbola whose general equation would be: The y = +/- .5x refers to the equation of the asymptotes for the hyperbola. In a horizontally oriented hyperbola, that equation would be: y = +/- In a vertically oriented hyperbola, that equation would be: y = +/- The distance between the center of the hyperbola and each vertex of the hyperbola is equal to "a". Since your center is (-2,0) and one of your vertices is at (2,3), then "a" must be equal to 3. Since the general equation of the asymptotes of a vertically oriented hyperbola is: y = +/- and the equation of the asymptotes of your equation is y = +/- , then the equation of the asymptotes for your hyperbola becomes: y = +/- .5x = +/- This makes: +/- .5x = +/- The slope of the asymptote on the left side of this equation is +/- .5 The slope of the asymptote on the right side of this equation is +/- . Since a = +/- 3, this makes the slope of the asymptote on the right side of this equation equal to Because these slopes are just different versions of the slope of the same asymptotes, we can solve for b as follows: +/- .5 = +/- Multiply both sides of this equation by b to get: +/- .5b = +/- 3 Multiply both sides of this equation by 2 to get: +/- b = +/- 6 We now have: a = 3 b = 6 Since the general form of the equation of our vertically oriented hyperbola is: , we can replace a^2 with 9 and b^2 with 36 to get: Since the center of our hyperbola is at (-2,0), and the center of the hyperbola is represented by (h,k), this means that: h = -2 k = 0 We can replace h and k in the equation of our hyperbola to get: That's the equation of our hyperbola. To graph this equation, we have to solve for y. Solving for y gets us: y = +/- The graph of our equation looks like this: A more distant view looks like this: The equation for the asymptotes of this hyperbola are y = +.- .5 We add the equations of those lines to the distant graph to get: The asymptotes are a little off because we did not account for the center of the hyperbola. Apparently, the equation y = +/- .5x was the equation for the slope of the asymptotes only. Since the asymptotes have to go through the center of the hyperbola, we need to modify this equation to account for the fact that the lines are going through the point (-2,0). The point slope form of the equation of a straight line is (y-y1) = m*(x-x1) The slope of our line is +/- .5 The point slope form of the equation of the asymptotes of our hyperbola becomes: (y-y1) = +/- .5*(x-x1) Since the lines goes through the point (-2,0), then (x1,y1) = (-2,0) and the equation of our asymptotes becomes: y = +/- .5*(x+2) We graph these equations of our asymptotes on top of the equation of our hyperbola to get: Now everything lines up right and the graph clearly shows the asymptotes of the equation of our hyperbola. You did not need to graph the asymptotes. That's a bonus. Another bonus is the eccentricity of the hyperbola. It is given by the equation c is the measure of the distance from the center of the hyperbola to one of the foci of the hyperbola. The value of c is given by the equation In this hyperbola, that would make The eccentricity of this hyperbola is therefore = 2.236067978 The higher the eccentricity, the straighter is the hyperbola. A very very high eccentricity would make the hyperbola look more like two parallel lines. This one is somewhere in the medium range. "a" and "c" are defined as: a is the distance from the center of the hyperbola to each vertex. c is the distance from the center of the hyperbola to each focus. A pretty decent reference for you to look at is shown below: http://www.purplemath.com/modules/hyperbola.htm
 Numeric_Fractions/298195: How do I find the inverse of the following function? g(x) = -(1/5)x+11 solutions Answer 214585 by Theo(3464)   on 2010-04-29 07:40:34 (Show Source): You can put this solution on YOUR website!g(x) = -(1/5)x+1 You solve for x and then you invert x and y as follows: Let y = g(x). Equation becomes: y = -(1/5)*x + 1 Solve for x as follows: Subtract 1 from both sides of the equation to get: y - 1 = -(1/5)*x Multiply both sides of the equation by 5 to get: 5 * (y-1) = -x Multiply both sides of the equation by -1 to get: -5 * (y-1) = x Simplify to get: x = -5y + 5 Invert x and y in this equation to get: y = -5x + 5 That should be your inverse equation. Replace y with h(x) to get: h(x) = -5x + 5 You could have used any letter other than h(x). a(x), b(x), c(x) would have done just as well. I just chose h(x) arbitrarily because that letter wasn't used yet and it was the next letter in line in the alphabet. g(x) is your equation. h(x) is your inverse equation. If h(x) is the inverse equation, you should be able to see that h(x) and g(x) are reflections about the line y = x. Graph both of these equations plus equation of y = x to see if that's true. Graph is shown below: The line in the middle going from bottom left to top right is the line of y = x. The line with the very steep slope going from top left to bottom right is the line of y = -5x + 5. The line with the shallow slope going from top left to bottom right is the line of y = -(1/5)x + 1. It's a little hard top see but they look like they're reflections about the line y = x (mirror images). The other way to tell is because the inverse function undoes what the function does. What this results in is that h(g(x)) = x Here's how that works. g(x) = -(1/5)*x + 1 Let x = any real number. We'll choose 160. Replace x in g(x) with 160 to get: g(160) = -(1/5)*(160) + 1 which equals -31 h(x) = -5x + 5 Replace x in h(x) with (-31) that we just calculated for g(x) to get: h(g(x)) = h(-31) = -5*-31 + 5 which becomes 155 + 5 which equals 160. g(160) = -31 h(-31) = 160 h(x) undid what g(x) did which makes h(x) the inverse function of g(x). You could also have shown that h(g(x)) = x by just solving the equations in x without substituting numbers for x. Here's how you would have done that: g(x) = -(1/5)*x + 1 h(x)= -5x + 5 h(g(x)) = -5 * (-(1/5)*x + 1) + 5 You replaced x in h(x) with g(x) and you replaced x in (-5x + 5) with (-(1/5)*x + 1) Simplify by distributing the multiplication to get: h(g(x)) = (-5 * (-1/5) * x) - (5 * 1) + 5 Since -5 * (-1/5) * x) = x, and - (5 * 1) = - 5, your equation becomes: h(g(x)) = x - 5 + 5 Combine like terms to get: h(g(x)) = x That shows h(x) is the inverse function of g(x).
 Linear-equations/297579: My child is in pre-al. She is trying to complete an xtra credit project. She is , from my understanding, looking for a graph that would trick you. I guess she is saying if you stretch the graph vs. keeping it basically a box type graph. Do you see what I am saying? Can you help me or point me to where I could print something out. I pretty much do not know what I am looking for.1 solutions Answer 214279 by Theo(3464)   on 2010-04-28 06:00:07 (Show Source): You can put this solution on YOUR website!Here's a graph of a circle. Here's a graph of the same circle. In the first graph it looks like a circle. In the second graph it looks like an ellipse. All I did was change the scale of the vertical axis (the y-axis). In the first graph the x and y axis spanned from -10 to 10. In the second graph, the x axis spanned from -10 to 10 and the y axis spanned from -20 to 20. If you are not aware of the scaling of each axis, you can definitely be misled by what's on the graph.
 Triangles/297609: Use the Pythagorean theorem to find the length of the unknown side of the triangle. Write the answer as a radical in simplified form. Square root of 17 is one side. Square root of 5 is the second side. And x is the third side.1 solutions Answer 214277 by Theo(3464)   on 2010-04-28 05:52:29 (Show Source): You can put this solution on YOUR website!Pythagorean Theorem is c^2 = a^2 + b^2 c is the hypotenuse and a and b are the legs of the right triangle. The hypotenuse is always bigger than either of the 2 legs. You are given that one side of the triangle is equal to sqrt(17). You are also given that the second side of the triangle is equal to sqrt(5). sqrt(5) cannot be the hypotenuse of the right triangle. sqrt(17) could. sqrt(x) could also, since you do not know what size that is. Assuming that x is the hypotenuse of the triangle, then the Pythagorean formula would be: x^2 = 17 + 5 c^2 = x^2 a^2 = 17 b^2 = 5 Simplify to get: x^2 = 23 That would make x = sqrt(23). Assuming that 17 is the hypotenuse of the triangle, then the Pythagorean formula would be: 17 = 5 + x^2 c^2 = 17 a^2 = 5 b^2 = x^2 Subtract 5 from both sides of that equation to get: x^2 = 17 - 5 Combine like terms to get: x^2 = 12 That would make x = sqrt(12). Either answer would be correct if it was not specified which side was which. In a right triangle, the Pythagorean Formula holds: c^2 = a^2 + b^2 With our first triangle, this becomes: 5 + 17 = 23 With our second triangle, this becomes: 5 + 12 = 17 One of these will be your correct answer once you determine which side was supposed to be 5 and which side was supposed to be 17.
 Problems-with-consecutive-odd-even-integers/297621: What is the smallest positive integer which multiplied by 40 gives a perfect square? (A) 2 (B) 5 (C) 20 (D) 40 (E) None of these1 solutions Answer 214275 by Theo(3464)   on 2010-04-28 05:09:26 (Show Source): You can put this solution on YOUR website!A perfect square is an integer that is the result of the squaring of another integer. 25 would be a perfect square because it is the result of 5*5 169 would be a perfect square because it is the result of 13*13 the factors of the perfect square can be positive or negative. Example 5*5 = 25 (-5)*(-5) = 25 Both 5 and -5 are square roots of the perfect square of 25. In your problem: square root of (2*40) = square root of 80 = 8.94427191 (not an integer) square root of (5*40) = square root of 200 = 14.14213562 (not an integer) square root of (20*40) = square root of 800 = 28.28427125 (not an integer) square root of (40*40) = square root of 1600 = 40 (this is the perfect square) Answer is selection D.
 Quadratic_Equations/297565: Graph the quadratic function f(x) = -x2 + 1 Describe the correct graph and why...Which way does the parabola go up or down, where does the graph cross the x-axis and y-axis.... 1 solutions Answer 214273 by Theo(3464)   on 2010-04-28 04:50:12 (Show Source): You can put this solution on YOUR website!Graph the quadratic function f(x) = -x^2 + 1 Describe the correct graph and why...Which way does the parabola go up or down, where does the graph cross the x-axis and y-axis.... The graph points up and opens down. Standard form of quadratic equation is ax^2 + bx + c = 0 Set f(x) = to 0 and you have this equation in standard form. You get: -x^2 + 1 = 0 In this equation: a = -1 b = 0 c = 1 Maximum point is at x = -b/2a which becomes 0 When x = 0, y = 1, so the maximum point is (x,y) = (0,1). To find the points where this graph crosses the x-axis, you have to solve the equation -x^2 + 1 = 0 With this equation, you subtract 1 from both sides of the equation to get: -x^2 = -1 Multiply both sides of this equation by -1 to get: x^2 = 1 Take the square root of both sides of this equation to get: x = +/- 1 Those should be the x-axis crossing points. You could also have factored the equation of -x^2 + 1 = 0 to get: (-x+1) * (x+1) = 0 When either of these factors = 0, the equation is grue, so you set each of the factors equal to 0 and solve. You get: x = -1 and x = 1. You graph this equation by plotting some values of x and getting corresponding values of y. You start with x = -1, x = 0, x = 1 That should be enough to draw a rough graph, but you might want to fill in some additional points to fit the curve better. Your graph should look like this:
 Polynomials-and-rational-expressions/297077: (y^2-8)(4y^2-4y+7) Multiply 16b^6-2p^6 Factor (x^5-6)+(x^5+6) Add Please help, thank you1 solutions Answer 213985 by Theo(3464)   on 2010-04-27 07:54:17 (Show Source): You can put this solution on YOUR website!First problem: (y^2 - 8) * (4y^2 - 4y + 7) equals: y^2 * (4y^2 - 4y + 7) - 8 * (4y^2 - 4y + 7) which equals: 4y^4 - 4y^3 + 7y^2 - 32y^2 + 32y - 56 Combine like terms to get: 4y^4 - 4y^3 - 25y^2 + 32y - 56 Second problem: 16b^6 - 2p^6 You can factor out a 2 to get: 2 * (8b^6 - p^6) Third problem: (x^5 - 6) + (x^5 + 6) equals: x^5 - 6 + x^5 + 6 after removing parentheses. Combine like terms to get: 2x^5.
 Evaluation_Word_Problems/297072: Translate the problem into a pair of linear equations in two variables. Solve the equations using either elimination or substitution. State your answer for the specified variable. From a point on a river, two boats are driven in opposite directions, one at 10 miles per hour and the other at 8 miles per hour. In how many hours will they be 54 miles apart? (Points :5) 1 solutions Answer 213984 by Theo(3464)   on 2010-04-27 07:33:00 (Show Source): You can put this solution on YOUR website!Rate * Time = Distance D = 54 miles. Total distance is the distance traveled by the first boat plus the distance traveled by the second boat. The time traveled by both boats will be the same. We call it T. the distance traveled by both boats is 54 miles. The distance traveled by the first boat is equal to x (we assigned that). the distance traveled by the second boat is equal to y (we assigned that also). x + y = 54. The rate of the first boat is 10 miles an hour. the rate of the second boat is 8 miles an hour. We have 2 equations that need to be solved simultaneously. The first equation is: 10 * T = x the second equation is : 8 * T = y We also know that x + y = 54 Since x = 10 * T and y = 8 * T, we can susbtitute in the equation of x + y = 54 to get: 10 * T + 8 * T = 54 This means that 18 * T = 54 which means that T = 3. Substituting for T in our equation gets: 10 * 3 + 8 * 3 = 54 which becomes 30 + 24 = 54 which becomes 54 = 54 which is true. The boats will be 54 miles apart in 3 hours. The first boat will have traveled 30 miles in 3 hours at 10 miles per hour. The second boat will have traveled 24 miles in 3 hours at 8 miles per hour. Since they are going in opposite directions, the total distance between them will be 30 + 24 = 54 miles in 3 hours.
 Rate-of-work-word-problems/297131: Water pours into a container at a constant rate of 4 litres per minute. When there are 50 litres of water in the container, a pump begins to pump water out at a rate of 5 litres per minute. How many minutes will it take to empty the container? (A) 10 (B) 24 (C) 50 (D) 120 (E) None of these1 solutions Answer 213981 by Theo(3464)   on 2010-04-27 05:29:12 (Show Source): You can put this solution on YOUR website!Rate * Time = Units Rate of water pouring into the container is 4 liters per minute. Rate of water being pumped out of the container is 5 liters per minute. The net rate of water leaving the container is equivalent to 1 liter per minute. When there are 50 liters of water in the container, the equation becomes: 1 * Time = 50 Time = 50 minutes. It would take 50 minutes to empty the container assuming that water is pouring in at 4 liters per minute and water is being pumped out at 5 liters per minute. Look at it from the perspective of the actual rates and you'll see how this works. There are 50 liters of water in the tank. Water is pouring in at 4 liters per minute for the next 50 minutes. 50 * 4 = 200 liters on top of the 50 liters that are already in the tank making a total of 250 liters. In the same 50 minutes, the pump is taking water out of the container at 5 liters per minute. 50 * 5 = 250 liters of water being pumped out of the container. This leaves the container empty after 50 minutes since the same amount of water that was pouring in, plus the amount of water that was already in the containter, has been pumped out. The equation that we used was 1 * T = 50 which made T = 50. T represents Time. We could have used another equation as follows: 4*T + 50 = 5*T When the values are equal, the tank is empty because 4*T + 50 is the number of liters of water coming in (we start at 50), and 5*T is the number of liters of water going out. Solve for this equation to get: T = 50. Same answer. In 50 minutes, the tank will be empty.
 Travel_Word_Problems/297103: one plane flies at a ground speed 50 miles per hour faster than another. on a particular flight, the faster plane requires 3 hours and the slower one 3 hours and 30 minutes. what is the distance of the flight?1 solutions Answer 213980 by Theo(3464)   on 2010-04-27 05:14:30 (Show Source): You can put this solution on YOUR website!Speed of the slower plane = x miles per hour. Speed of the faster plane = (x+50) miles per hour. Rate * Time = Distance. Faster plane equation becomes: (x+50) * 3 = D Slower plane equation becomes: x * 3.5 = D Since they both equal D, then they both equal each other and we get: (x+50) * 3 = x * 3.5 Simplify to get: 3*x + 150 = 3.5*x Subtract 3*x from both sides of the equation to get: .5*x = 150 Divide both sides of the equation by .5 to get: x = 300 The slower plane is flying at 300 miles per hour. The faster plane is traveling at 350 miles per hour. the equation for the faster plane becomes: 350 * 3 = D = 1050 miles. The equation for the slower plane becomes: 300 * 3.5 = D = 1050 miles. the distance is the same in both equations as it should be. The distance is 1050 miles.
 Quadratic_Equations/297091: 16r^3-9r=0 solve please. thank you so much1 solutions Answer 213976 by Theo(3464)   on 2010-04-27 03:55:23 (Show Source): You can put this solution on YOUR website!16r^3 - 9r = 0 Factor out the r on the left side of the equation to get r * (16r^2 - 9) = 0 r = 0 will satisfy this equation because anything times 0 equals 0. 16r^2 - 9 = 0 will also satisfy this equation because anything times 0 equals 0. Solve for 16r^2 - 9 = 0 Add 9 to both sides of this equation to get 16r^2 = 9 Divide both sides of this equation by 16 to get r^2 = (9/16) Take the square root of both sides of this equation to get r = +/- sqrt(9/16) Since sqrt(9/16) is the same as sqrt(9) / sqrt(16), and since sqrt(9) = 3, and since sqrt(16) = 4, you get r = +/- 3/4 You have 3 possible answers to this equation. They are: r = 0 r = 3/4 r = -3/4 Plug those values into the original equation to see if they hold up. Your original equation is 16r^3 - 9r = 0 When r = 0, this equation becomes 0 - 0 = 0 which is true. When r = 3/4, this equation becomes 16*27/64 - 9*3/4 = 432/64 - 27/4 = 6.75 - 6.75 = 0 which is true. When r = -3/4, this equation becomes 16*-27/64 + 9*3/4 = -432/64 + 27/4 = -6.75 + 6.75 = 0 which is true. All 3 answers are solutions to the original equation.
 Miscellaneous_Word_Problems/297102: I need help please. I need to write an inequality and explain the answer. If I need 30 tons of rock to cover an area and each ton costs \$60.00 and each tree is \$84.00, what is the maximum number of trees I can buy with \$2500.00? Would 5 trees be a solution to the inequality? Please advise how I would write this. Thank you.1 solutions Answer 213974 by Theo(3464)   on 2010-04-27 03:38:38 (Show Source): You can put this solution on YOUR website!You need 30 tons of rock to cover the area. That is fixed. 30 * 60 = \$1800. You have \$2500 to spend. \$2500 - 1800 = \$700. You have \$700 to spend on trees. \$700 / \$84 = 8.333333333 trees. You can buy up to 8 trees. the formula you would use is as follows: Let T = number of trees. Let R = number tons of rocks. Let x = price per ton of rocks. Let y = price per tree. the general formula you would use is T <= When x = \$60 and R = 30 and y = \$84, this formula becomes: T <= Simplify this equation to get: T <= Simplify further to get: T <= Simplify further to get: T <= 8.333333333 Since you can't buy a partial tree, you can buy up to 8 trees. "<=" means less than or equal to
 Probability-and-statistics/296701: A) A test consists of 10 true and false questions. To pass the test a student must answer at least eight questions correctly. If the student guesses on each questions what is the probability that the student will pass the test?1 solutions Answer 213773 by Theo(3464)   on 2010-04-26 13:45:50 (Show Source): You can put this solution on YOUR website!Assume it was only 3 answers on the test and the student needed to get at least 2 out of 3 correct. Probability of getting exactly 0 wrong would be .5^3 = .125 * 1 = .125 Probability of getting exactly 1 wrong would be .5^3 = .125 * 3 = .375 Probability of getting exactly 2 wrong would be .5^3 = .125 * 3 = .375 Probability of getting exactly 3 wrong would be .5^3 = .125 * 1 = .125 Total probability is equal to 1 as it should be. Probability of getting 0 or 1 wrong would be .375 + .125 = .5 Since 0 or 1 wrong is the same as getting 2 or 3 right, then this is the probability that the student will get at least 2 right. The individual probabilities are multiplied by the number of ways they can occur. If we let 0 = wrong and 1 = correct, then: You can get 0 wrong only 1 way (111) You can get 1 wrong 3 ways (110) (101) (011) You can get 2 wrong 3 ways (001) (010) (100) You can get 3 wrong 1 way (000) The same concept applies to the larger numbers. ----- With 10 answers, this is what happens: p(0) = probability of getting exactly 0 correct. p(1) = probability of getting exactly 1 correct. etc. p(0) = .5^10 = .000976563 * 1 = .000976563 p(1) = .5^10 = .000976563 * 10 = .009765625 p(2) = .5^10 = .000976563 * 45 = .043945313 p(3) = .5^10 = .000976563 * 120 = .1171875 p(4) = .5^10 = .000976563 * 210 = .205078125 p(5) = .5^10 = .000976563 * 252 = .24609375 p(6) = .5^10 = .000976563 * 210 = .205078125 p(7) = .5^10 = .000976563 * 120 = .1171875 p(8) = .5^10 = .000976563 * 45 = .043945313 p(9) = .5^10 = .000976563 * 10 = .009765625 p(10) = .5^10 = .000976563 * 1 = .000976563 Total probability equals 1 as it should. Probability of getting 0 or 1 or 2 wrong is equal to: .000976563 + .009765625 + .043945313 = .0546875 Since the probability of getting 0 or 1 or 2 wrong is the same as the probability of getting 8 or 9 or 10 right, then the probability that the student will get at least 8 correct is equal to .0546875. The number of ways each percentage can be achieved is given by the formula: For example, with 10 answers, the number of ways of getting exactly 4 wrong is equal to: With 10 answers, the number of ways of getting exactly 6 wrong is the same as getting exactly 4 wrong as shown below:
 Quadratic_Equations/296203: I need help translating the problem situation to a system of equations. Sarah Comar's Candy Store sold a total of 53 pounds of jelly beans, selling two kinds of jelly beans. The first kind was priced at \$4.45 pound, and the second kind was priced at \$1.12 per pound. In all, \$125.96 was taken in for the two types of jelly beans. How many pounds of each kind were sold? (Let x represent the number of pounds of the first kind and y represent the number of pounds of the second. 1 solutions Answer 213495 by Theo(3464)   on 2010-04-25 08:16:15 (Show Source): You can put this solution on YOUR website!Your system of equations would be as follows: x = the number of pounds of the first kind of jelly beans. y = the number of pounds of the second kind of jelly beans. Total of 53 pounds of jelly beans was sold. Equation 1 would be: x + y = 53 The first kind of jelly beans was priced at \$4.45 per pound. The second kind of jelly beans was priced at \$1.12 per pound. The total amount of money made was equal to \$125.96 Equation 2 would be: 4.45 * x + 1.12 * y = 125.96 The two equations you have to solve simultaneously are: x + y = 53 (equation 1) 4.45 * x + 1.12 * y = 125.96 (equation 2) You can solve by substitution or by elimination. Either way will get you the same answer.
 Percentage-and-ratio-word-problems/296195: In an examination it is required to get 45% marks to pass. A student got 138 marks and failed by 15% of the total marks. What were the maximum marks? I know ans is 460 but plz help me1 solutions Answer 213483 by Theo(3464)   on 2010-04-25 06:39:50 (Show Source): You can put this solution on YOUR website!The student was required to get 45% of the maximum marks to pass. The student failed by 15% of the maximum marks. This means that the student got 45% of the maximum marks minus 15% of the maximum marks which equals 30% of the maximum marks. If we let M equal the maximum marks that the student can get, then the equation would be: Student got: .45*M - .15*M = (.45 - .15)*M = .30*M. Since the student got a score of 138 marks, then our equation becomes: .30*M = 138 Divide both sides of this equation by .30 to get: M = 138/.3 Solve for M to get M = 460.
 Exponents-negative-and-fractional/296180: simplify 3k^4 x m^-31 solutions Answer 213481 by Theo(3464)   on 2010-04-25 06:24:21 (Show Source):
 Angles/296176: what is the largest angle of a triangle where one is 8 times greater than other. 1 solutions Answer 213480 by Theo(3464)   on 2010-04-25 05:56:54 (Show Source): You can put this solution on YOUR website!The largest angle has to be smaller than 160 degrees. If the largest angle is 160 degrees, then the angle that it is 8 times larger than is equal to 160/8 = 20 degrees. The sum of these 2 angles is 180 degrees which forces the third angle to be 0 degrees which is not allowed since each angle of the triangle has to be greater than 0 degrees. So the largest angle has to be less than 160 degrees. You will never be able to find the largest angle because, no matter how close you can get to 160 degrees, you will always be able to get a little closer. It's simpler to say that the largest angle has to be smaller than 160 degrees.