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# Recent problems solved by 'Theo'

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 Miscellaneous_Word_Problems/284291: The literature club is printing a storybook to raise money. The print shop charges \$3 for each book, and \$30 to create the film. How many books can the club print if their budget is \$705? a)223 b) 235 c) 237 d) 225 e) 245 1 solutions Answer 206286 by Theo(3464)   on 2010-03-23 05:00:26 (Show Source): You can put this solution on YOUR website!Assuming that they are going to buy 1 film, then the amount of money they have left for books is \$705 minus \$30 = \$675. \$675 / \$3 = 225. They will be able to buy 225 books with the money they have left over after they buy the film.
 Quadratic-relations-and-conic-sections/283837: what is the equation of the ellipse with foci at (0,-4) and (0,4) and the sum of its focal radii being 10. this confuses me! please help if possible thanks1 solutions Answer 206111 by Theo(3464)   on 2010-03-22 16:27:27 (Show Source): You can put this solution on YOUR website!An ellipse has a major axis and a minor axis. In your equation, the major axis is horizontal and the minor axis is vertical. The foci are at (0,-4) and (0,4). The center of the ellipse is at the point (0,0). c is the distance between each focus and the center of the ellipse. This means that c = +/- 4. This means that c^2 = 16 The major axis of the ellipse is equal to 2a. This means that a is the distance from the center of the ellipse to the intersection of the major axis of the ellipse with the ellipse. The minor axis of the ellipse is equal to 2b. This means that b is the distance from the center of the ellipse to the intersection of the minor axis of the ellipse with the ellipse. the major axis of this ellipse is vertical. the minor axis of this ellipse is horizontal. A diagram of the major axis of this ellipse and the minor axis of this ellipse and the location of the foci and the intersection of the major and minor axis of this ellipse with the ellipse is shown below. everything is drawn except the ellipse itself which I can't do. ``` P(major) x F2 b x | x | x (minor)P x x x C x x x P(minor) x x a ----- x x ----- c x x F1 x P(major) ``` In the diagram above: P(minor) is the intersection of the minor axis with the ellipse. P(major) is the intersection of the major axis with the ellipse. F1 is the focal point at (0,-4). F2 is the focal point at (0,4). C is the center of the ellipse at (0,0). c is the straight line distance between C and F1. a is the straight line distance between F1 and P(minor) on the left of the diagram (line not shown). b is the straight line distance between C and P(minor) on the left of the diagram. This forms a right triangle with the points of the triangle being F1, C, and P(minor) on the left of the diagram. By the Pythagorean formula, a^2 = b^2 + c^2. This formula leads to c^2 = a^2 - b^2. Since c^2 = a^2 - b^2, and since c = 4, this means that: 16 = a^2 - b^2 The problem states that the sum of the focal radii = 10. The sum of the focal radii is a constant and is equal to the distance between F1 and a point on the ellipse plus the distance between F2 and the same point on the ellipse. By definition, the sum of these distances is equal to 2a which is the diameter of the major axis. Since 10 is equal to 2a, this means that a = 5 which is the length from the center of the ellipse to the perimeter of the ellipse along the major axis. In the diagram, that would be from C to P(major) on the top of the diagram. In the diagram, it is also from C to P(major) on the bottom of the diagram. Since we know that a = 5, and we know that c = 4, we can derive b using the formula: c^2 = a^2 - b^2 This becomes 4^2 = 5^2 - b^2 which becomes: 16 = 25 - b^2. We solve for b to get b = sqrt(9) = 3. b is the length from the center of the ellipse to the ellipse along the minor axis. In the diagram, that would be from C to P(minor) on the left of the diagram. In the diagram, that would also be from C to P(minor) on the right side of the diagram. We now have: a = 5 b = 3 c = 4 From this, we should be able to derive the formula for the ellipse. The standard formula for an ellipse is: (x-h)^2 / b^2 + (y-k)^2 / a^2 = 1 Since the two foci are at (0,-4) and at (0,4), this means that the vertex of the ellipse (otherwise known as the center of the ellipse) is at (0,0). This means that (h,k) = (0,0), because (h,k) represent the center of the ellipse (otherwise known as the vertex of the ellipse). Since h is zero, then (x-h)^2 becomes x^2. since k is zero, then (y-k)^2 becomes y^2. Our formula becomes: x^2 / b^2 + y^2 / a^2 = 1 Since a = 5 and b = 3, this formula becomes: x^2 / 3^2 + y^2 / 5^2 = 1 which becomes: x^2 / 9 + y^2 / 25 = 1 To graph this equation, we have to solve for y as follows: Equation is: x^2 / 9 + y^2 / 25 = 1 Subtract x^2 / 9 from both sides of the equation to get: y^2 / 25 = -x^2/9 + 1 Multiply both sides of the equation by 25 to get: y^2 = (-25/9)x^2 + 25 Take the square root of both sides of the equation to get: y = +/- sqrt((-25/9)x^2 + 25) This makes y = sqrt((-25/9)x^2 + 25) and y = -sqrt((-25/9)x^2 + 25) Graph of this equation looks like this: In this graph, the foci are at (x,y) = (0,-4) and at (x,y) = (0,4). The major axis is vertical at y = 0. The minor axis is horizontal at x = 0. The vertex is at (x,y) = (0,0). The distance from the foci to the vertex is +/- 4 along the y-axis. The sum of the focal radii will always be 10. The answer to your question is that the equation for this ellipse is: x^2/9 + y^2/25 = 1 The general form of that equation is: x^2/b^2 + y^2/a^2 = 1 A picture of this ellipse is shown below: In this picture: 2a is the length of the major axis. The designation for the distance from the surface of the ellipse to the center of the ellipse along the major axis is called "a". 2b is the length of the minor axis. The designation for the distance from the surface of the ellipse to the center of the ellipse along the minor axis is called "b". The designation for the distance from each focal point to the center of the ellipse is called "c". F1 and F2 are the focal points of the ellipse. The center of this ellipse is (0,0). F1 is at (0,4) and F2 is at (0,-4). ***** a also happens to be the hypotenuse of the right triangle formed by b and c as shown in the diagram. This is why the formula a^2 = b^2 + c^2 is valid. Since a^2 = c^2 + b^2 is a valid equation (Pythagorean Formula for Right Triangle), the other formula that can be derived from this is: c^2 = b^2 - a^2. ***** What is not shown in this picture is the focal radii. That will be shown in a separate picture below: In this picture: P1 is a point on the surface of the ellipse. P2 is a point on the surface of the ellipse. d11 is the distance from P1 to F1. d12 is the distance from P1 to F2. d21 is the distance from P2 to F1. d22 is the distance from P2 to F2. The focal radii shown on this ellipse are d11, d12, d21, d22. From P1, the Sum of the focal radii are d11 and d12. From P2, the Sum of the focal radii are d21 and d22. The Sum of the focal radii is a contant. This means that the sum will always be the same, regardless of which point on the surface of the ellipse you are at. The Sum of d11 and d12 is equal to 10 for this ellipse. The Sum of d21 and d22 is also equal to 10 for this ellipse. A fairly decent reference regarding this problem can be found at the following website. http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_ellipse.xml
 Surface-area/283829: Why would someone need to calculate the Surface Area of the Pyramids?1 solutions Answer 205974 by Theo(3464)   on 2010-03-22 07:47:18 (Show Source): You can put this solution on YOUR website!I don't really know. But, ..... I did a search on the internet and found an article that indicated that the surface area of the pyramids was originally limestone. Maybe they needed to determine how much limestone they needed? Maybe somebody today wanted to know how much limestone they needed? Here's the article. http://arc.boardofstudies.nsw.edu.au/go/sc/maths/activities/the-pyramid-of-giza/
 Geometry_Word_Problems/283823: If a circle of radius 2 cm is inscribed in a square, what is the area of the square? a)4 cm2 b) 4 π cm2 c) 8 cm2 d) 16 cm2 e) 16 π cm2 1 solutions Answer 205969 by Theo(3464)   on 2010-03-22 07:08:08 (Show Source): You can put this solution on YOUR website!I would say 16cm^2. The radius of the circle is equal to 2cm. The diameter of the circle is equal to 2 times the radius is equal to 4cm. The diameter of the circle is equal to one side of the square since the circle is inscribed in the square. The area of the square is equal to s^2 is equal to 4^2 is equal to 16cm^2. Answer is selection d.
 logarithm/283763: Express as a sum of logarithms. log[4](64 ∙ 256)1 solutions Answer 205960 by Theo(3464)   on 2010-03-22 05:00:44 (Show Source): You can put this solution on YOUR website!In algebgra.com language, the answer is: log(4,(64*256)) = log(4,64) + log(4,256) After being put through the algebra.com formula generator, the answer looks like this: In english, the answer is: log(64*256) to the base of 4 is equal to log(64) to the base of 4 plus log(256) to the base of 4. In your language, the answer is: log[4](64*256) = log[4](64) + log[4](256). This is because, in general: log(b,a*c) = log(b,a) + log(b,c)
 Rational-functions/283760: graph x^2-8x+21 solutions Answer 205958 by Theo(3464)   on 2010-03-22 04:52:57 (Show Source): You can put this solution on YOUR website!graph of y = x^2 - 8x + 2 is shown below: To plot the points on this graph, all you do is take values of x and then solve for y assuming that value of x replaces x in the equation. example: when x = 4, y = 4^2 - 8*4 + 2 = 16 - 32 + 2 = -16 + 2 = -14.
 Expressions-with-variables/283789: simplify each complex fraction ( 2/9 +4/9 ) (1/3 - 9/10)1 solutions Answer 205957 by Theo(3464)   on 2010-03-22 04:44:15 (Show Source): You can put this solution on YOUR website!expression is: (2/9 + 4/9) * (1/3 - 9/10) 2/9 + 4/9 can be combined because their denominators are the same to get: 6/9 1/3 - 9/10 needs to find a common denominator for. easiest one to find is 3 * 10 = 30. multiply first fraction by 10/10 and multiply the second fraction by 3/3 to get: 10/30 - 27/30. now that the denominator are the same, these can be combined to equal: -17/30. your expression of (2/9 + 4/9) * (1/3 - 9/10) is now equivalent to: (6/9) * (-17/30) which can be multiplied together to get: (6*-17)/(9*30) = -102/270 -102/270 = -.3777777778 Use your calculator to solve the original expression to see if it is the same as the result you get with the final expression. They are the same so the simplification process was correct. If you did not want to multiply the first expression by the second expression, then your answer would be: first expression: (2/9 + 4/9) is the same as 6/9 = 2/3 second expression: (1/3 - 9/10) = (10/30 - 27/30) = -17/30
 Expressions-with-variables/283790: simplify each complex fraction 2 x (1/4 + 1/5) + 21 solutions Answer 205956 by Theo(3464)   on 2010-03-22 04:34:54 (Show Source): You can put this solution on YOUR website!2 * (1/4 + 1/5) + 2 1/4 is the same as 5/20 1/5 is the same as 4/20 expression becomes: 2 * (5/20 + 4/20) + 2 since the denominators are now the same, you can combine the fractions to get: 2 * (9/20) + 2 2 * 9/20 = 18/20 so your expression becomes: 18/20 + 2 18/20 can be simplified to 9/10 so your expression becomes: 2 and 9/10 2 is equivalent to 20/10. your expression becomes 20/10 + 9/10. Since the denominator is the same, these fractions can be combined to equal: 29/10 which equals 2.9 You can confirm this is accurate by using your calculator to determine the value of the original expression and see if it is the same as the value of the final expression. I confirmed and they are the same so the answer is good.
 Money_Word_Problems/283803: Good day. I need some help on this one problem, and I'd be terribly appreciative if anyone would bother to check it out. Here goes: A man insured a building for \$10,000 at an annual rate of 1-1/2%. He equipped the building with automatic sprinklers at a cost of \$1,250. The fire insurance company then reduced the rate on his property to 1/4% per year. In how many years would the savings on the premiums pay for the sprinklers? Thank you very much and more power!1 solutions Answer 205955 by Theo(3464)   on 2010-03-22 04:23:27 (Show Source): You can put this solution on YOUR website!man insures building for \$10,000 at annual rate of 1 and 1/2 percent per year. That's 1.5% divided by 100% = a rate of .015 per year. This means he would be paying .015*10000 = 150 per year. He pays \$1250 for a sprinkler. Insurance company reduces the rate on his property to 1/4% per year. That's .25% divided by 100% = a rate of .0025 per year. This means he would be paying .0025*10000 = 25 per year. The difference in what he would be paying per year would be 150 - 25 = 125 a year. In other words, he would be saving 125 per year on the insurance rates he is paying because he bought the sprinkler. 1250 / 125 = 10. He will make his money back in 10 years.
 logarithm/283809: 2log(7,3) + 3log(7,2) = log(7, x)1 solutions Answer 205953 by Theo(3464)   on 2010-03-22 03:48:51 (Show Source): You can put this solution on YOUR website!2*log(7,3) + 3*log(7,2) = log(7,x) We can rewrite this as: log(7,x) = 2*log(7,3) + 3*log(7,2) The rules of logarithms that apply here will be: log(a^n) = n*log(a) log(a*b) = log(a) + log(b) First thing we notice is that 2*log(7,3) would be the same as log(7,3^2) which would be the same as log(7,9). Second thing we notice is that 3*log(7,2) would be the same as log(7,2^3) which would be the same as log(7,8). Our equation becomes: log(7,x) = log(7,9) + log(7,8) Third thing we notice is that log(7,9) + log(7,8) would be the same as log(7,9*8) which would be the same as log(7,72) Our equation becomes: log(7,x) = log(7,72) This can only be true if x = 72. Our original equation is: log(7,x) = 2*log(7,3) + 3*log(7,2) We replace x with 72 and we get: log(7,72) = 2*log(7,3) + 3*log(7,2) We can convert this equation to the base of 10 so we can solve using our calculator. The conversion formula in general is: log(b,x) = log(c,x)/log(c,b) This reads as log of x to the base b = log of x to the base c divided by the log of b to the base c. The conversion formula in our case is: log(7,x) = log(10,x) / log(10,7) Our equation becomes: log(10,72/log(10,7) = 2*log(10,3)/log(10,7) + 3*log(10,2)/log(10,7) If we multiply both sides of this equation by log(10,7), then we get: log(10,72) = 2*log(10,3) + 3*log(10,2) which becomes: 1.857332496 = 2*.477121255 + 3*.301029996 which becomes: 1.857332496 = 1.857332496 which is true confirming the equations are equivalent. Your answer is that x = 72.
 Money_Word_Problems/283808: Hello! I would be thankful if someone is able to help me out on this one problem of mine which is, quite frankly, totally beyond me, and it deals with budgets and such. It goes something like this: During the first 3 months of the fiscal year an operating department spent 90%, 105%, and 120% respectively of its monthly budget which remained the same each month. If the budegeted amount totalled \$252,000 for these three months, how much were expenditures below the budget estimates during the first month? Thank you very much for the help! :) 1 solutions Answer 205952 by Theo(3464)   on 2010-03-22 03:32:17 (Show Source): You can put this solution on YOUR website!If the total budget for the 3 months was \$252,000, and the monthly budget remained the same for the 3 months, then the monthly budget had to be \$252,000 / 3 = \$84,000 per month. If they spent 90% of their monthly budget the first month, then they spent .9 * \$84,000 = \$75,600 the first month. \$84,000 - \$75,600 = \$8,400. They spent \$8,400 below their budget for the first month.
 Geometry_Word_Problems/283298: hi, my question is following A toy rocket is launched straight up from the top of a building 50 ft tall at an initial velocity of 200 ft per sec. Using the function V(x)=-16t^2+vt+h, answer the following. 1. Give the function that describes the height of the rocket in terms of t? 2. Determine the time at which the rocket reaches its maximum height, and its maximum height in feet? 3. For what time interval will the rocket be more than 300 ft above the ground level? 4. for how many seconds will it hit the ground?1 solutions Answer 205718 by Theo(3464)   on 2010-03-21 11:00:28 (Show Source): You can put this solution on YOUR website!let x = t. your equation becomes V(x) = -16x^2 + vt + h h = 50 which is the height of the rocket at time point 0 (value of x is 0). your equation becomes v(x) = -16x^2 + vx + 50 Since initial velocity is 200 ft/sec, then v = 200 and your equation becomes: v(x) = -16x^2 + 200x + 50 maximum height should be when x = -b/2a a = -16 b = 200 c = 50 maximum height should be when x = -200/-32 = 6.25 when x = 6.25, v(x) = 675 feet. That's what the maximum height should be. Set v(x) = 300 and solve for x. 300 = -16x^2 + 200x + 50 subtract 50 from both sides of this equation to get 0 = -16x^2 + 200x - 250. solve for the roots of this equation by quadratic formula. a = -16 b = 200 c = -250 (-b +/- sqrt(b^2 - 4ac))/(2a) =: (-200 +/- sqrt(200^2 - 4*-16*-200))/(-32) =: (-200 +/- sqrt(40000 - 12800))/(-32) =: (-200 +/- sqrt(27200)))=32 =: (-200 +/- 164.924225)/(-32) x = 1.096117968 or x = 11.40388203 The rocket should be at or above 300 feet from x = 1.096117968 to x = 11.40388203 A graph of this equation is shown below: A horizontal line was placed at y = 300 and 675 to show you the intersection points with the graph of the equation of the rocket. At y = 675, the graph should be at the maximum point. At 300, the graph should be at x = 1.09 and x = 11.4 roughly. Trace a vertical line down from those intersection points to see the x-value. x-value represents time in seconds. y-value represents height in feet.
 Miscellaneous_Word_Problems/283360: How far will a wheel with radius 5/π roll in two revolutions? a) 25/π units b) 10 units c) 20 units d) none of these1 solutions Answer 205716 by Theo(3464)   on 2010-03-21 10:33:33 (Show Source): You can put this solution on YOUR website!circumference = 2*pi*r circumference = 1 revolution 1 revolution = 2*pi*r r = 5/pi 1 revolution = 2*pi*5/pi = 2*5 = 10 2 revolutions = 20 Answer is selection c.
 Signed-numbers/283384: -9-5=1 solutions Answer 205715 by Theo(3464)   on 2010-03-21 10:30:39 (Show Source): You can put this solution on YOUR website!-14
 Percentage-and-ratio-word-problems/283357: In a store, a \$100 item was marked down by 20% for a sale. After the sale, the itemʼs sale price was marked up 20%. What was the final price? a) \$96 b) \$100 c) \$64 d) \$1201 solutions Answer 205713 by Theo(3464)   on 2010-03-21 10:29:31 (Show Source): You can put this solution on YOUR website!100 - .2*100 = 100 - 20 = 80 80 + .2*80 = 80 + 16 = 96 Answer is selection a.
 Miscellaneous_Word_Problems/283351: A glass is half full of pure alcohol. A second glass whose volume is twice the volume of the first glass is one-third full of pure alcohol. Both glasses are then filled to the top with water and mixed together in a third container. What fraction of the final mixture is alcohol? A 5/6 b 5/12 c 7/12 d 5/18 e 7/18 1 solutions Answer 205710 by Theo(3464)   on 2010-03-21 10:22:58 (Show Source): You can put this solution on YOUR website!let the volume of the first glass be x let the volume of the second glass be 2x the volume of the container is 3x. The amount of alcohol in the first glass is (1/2)*x The amount of alcohol in the second glass is (1/3)*2x = (2/3)*x The total amount of alcohol in the container is (1/2)*x + (2/3)*x = (3/6)*x + (4/6)*x = (7/6)*x The total volume of the container is 3*x. The fraction of alcohol in the container is therefore (7/6)*x / (3*x) which is equal to (7/18). As an example: Assume the first glass is 6 ounces. It is half full of alcohol so this contains 3 ounces of alcohol. The second glass is twice the volume of the first glass so the volume of the second glass is 12 ounces. It is 1/3 full of alcohol so this contains 4 ounces of alcohol. The total alcohol in both glasses is 3 ounces + 4 ounces = 7 ounces. The total volume of the mixture in the container is 6 ounces + 12 ounces = 18 ounces. The fraction of alcohol in the container is 7 ounces / 18 ounces = 7/18