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 Circles/707548: find the equation of the circle inscribed in a triangle whose sides are 3x+y-5=0, x+3y-1=0, x-3y+7=0. 1 solutions Answer 435842 by KMST(1863)   on 2013-01-29 13:10:59 (Show Source): You can put this solution on YOUR website!The radius of the circle is the distance, , from center (h,k) to each of the lines. The distance from a point (h,k) to a line can be calculated as , so which simplifies to and With those absolute values, there are too many possibilities, so we should figure out the sign of the expressions inside the absolute values. It's easy to sketch the triangle and figure out. With the line equations in that form, it's easy to calculate intercepts: For example, wit , --> --> and --> --> --> So, goes through (5/3,0) and (0,5). similarly, goes through (1,0) and (0,1/3), and goes through (-7,0) and (0,7/3). Sketching, we can find what points are inside the triangle, including point (h,k), the center of the circle. The triangle is the space below the lines that go through (0,5) and (-7,0) (the first and third lines), but above the line for (the with intercepts so close to the origin. Each of the expressions inside the absolute values would be zero if (h,k) is on the corresponding line, positive if (h,k) is on one side of the line, and negative if it is on the other side. Since none of the lines goes through the origin, we can use the origin as a test point to figure out which side is which. is true for the part of the plane below , where we find the origin, (0,0), the triangle, and (inside the triangle) the center of the circle (h,k). So, and is true for the origin and the part of the plane below , but for the other side, where the triangle and (h,k) are. So, and is true for the origin and the part of the plane below , where we find the origin, (0,0), the triangle, and point (h,k). So, and Now that we know the sign of each expression, we can re-write as and that is a system of equations that we can solve to get , and --> --> Then we can write the equation of the circle as
 Angles/707556: If an angle is 100 degrees is it complementary or supplementary1 solutions Answer 435774 by KMST(1863)   on 2013-01-29 06:59:00 (Show Source): You can put this solution on YOUR website!An angle measuring 100 degrees would be supplementary to any angle measuring 80 degrees, but it cannot be called supplementary by itself. "Complementary" and "supplementary" are terms that describe the relationship between two angles. They are comparative words like "larger." Those words are not applied to an angle by itself, unless it is understood what other angle it is being compared to. Two angles are called supplementary (to each other) if they add up to 180 degrees. Two angles are called complementary (to each other) if they add up to a right angle (90 degrees). Unless you are older than I think, and your teacher has talked about negative angles, there is no angle that can be complementary with a 100 degree angle.
 Equations/707595: Please help me solve this equation: x^3-2012^y=26 x,y=?1 solutions Answer 435773 by KMST(1863)   on 2013-01-29 06:18:45 (Show Source): You can put this solution on YOUR website!With no other restrictions, there should be an infinity number of (x,y) pairs that are solutions. If x and y must be non-negative integers, (3,0) is the only solution I see.
 logarithm/707608: let suppose if we denote x as a length of any object in meter, than after taking log of x, is there any change in its unit's.1 solutions Answer 435772 by KMST(1863)   on 2013-01-29 06:07:54 (Show Source): You can put this solution on YOUR website!I would say the units for log(x) deserve a new name, because you cannot add the value of log(x) to lengths in meters, just as you cannot add any quantity in any other unit to lengths in meters. You could call it log-meter. However, I have never heard of a name for such a unit. The only logs that "sort of" got a name are 1 unit of pH = -log(1 mole of hydrogen ion per liter) and decibels. They are not exactly logs, but they involve logs.
 Inequalities/707611: hello there! I have a problem here. I know the answer but I dont know how to show the solution. Here is the problem: 3/2 < 3x/2 + 3 I think the answer is: x > -1 thank you!1 solutions Answer 435771 by KMST(1863)   on 2013-01-29 05:53:53 (Show Source): You can put this solution on YOUR website!I would get rid of denominators first by multiplying both sides of the inequality times 2: --> Next, I would subtract 6 from both sides: --> Finally, I would divide both sides by 3: -->
 Surface-area/707114: what is the area of a triangle with side that are 9 units, 17 units and 10 units? 1 solutions Answer 435729 by KMST(1863)   on 2013-01-28 22:33:30 (Show Source): You can put this solution on YOUR website!I say that the area is and here is the triangle: The small triangle, with sides measuring 6, 8, and 10 units is a right triangle, because the squares of the lengths of the shorter sides add up to the square of the length of the longer side: A large triangle with sides measuring 15, 8, and 17 units is also a right triangle, because the squares of the lengths of the shorter sides add up to the square of the length of the longer side: Our triangle with sides measuring 9, 10, and 17 units is the large right triangle minus the small right triangle, and its area is the difference between the areas of the two right triangles. Area of the large right triangle= square units Area of the small right triangle= square units Area of the triangle in the problem= square units When you only have the lengths of the sides, unless it is a right triangle, the problem is tricky. If you have not been taught enough, you need a combination of luck, ingenuity and triangle construction to figure out what I show in the drawing an explanation above. To construct a triangle with sides measuring 9, 10, and 17 cm, you would draw one of the sides to exact measure, and the use a compass to mark arcs centered at the ends of the first side, with radii matching the other two side lengths. Where the arcs meet is the third vertex of the triangle. Such a construction would help you figure out what I showed above. If you have been taught about Heron's formula involving the semi-perimeter, you could use that. If you were taught the law of cosines, you could use it to find the cosine (and then sine) of one angle, and then use the formula for area involving the lengths of two sides and the sine of the angle in between. USING LAW OF COSINES: units = measure of shortest side units = measure of medium length side units = measure of longest side =measure of the largest angle, opposite longest. The law of cosines says that so --> Solving for : so The area of a triangle can be calculated from the lengths of 2 sides, a and b, and the angle between them, C, as so USING HERON'S FORMULA: The semi-perimeter, , is defined as half of the perimeter. If the side lengths are , , and , then , and the area of the triangle is and
 Numbers_Word_Problems/706884: The sum of the squares of two consecutive even integers, the first of which is 2n. What do I do?1 solutions Answer 435713 by KMST(1863)   on 2013-01-28 21:05:08 (Show Source): You can put this solution on YOUR website!probably the question was to write an expression for that in algebra (not in words) and to simplify that expression. = the first of two consecutive even integers. = the next of the two consecutive even integers. The sum of their squares is
 Polynomials-and-rational-expressions/706870: for the following expression, what are the restrictions for a? 2a^2+a-15 over 5 a^2+16a+3 (1 point) 1 solutions Answer 435710 by KMST(1863)   on 2013-01-28 20:58:13 (Show Source): You can put this solution on YOUR website!The rational expression does not exist when which happens for and for (You can find that by factoring , or by solving using the quadratic formula). Those are the only restrictions on the variable . For any value of other than and the expression is defined.
 Polynomials-and-rational-expressions/706845: Factor the polynomial 18a^2b^3c-45ab^6c completely1 solutions Answer 435706 by KMST(1863)   on 2013-01-28 20:45:58 (Show Source): You can put this solution on YOUR website! has just two terms: and The numeric coefficients have as a common factor: and The variable appears as in one term and as in the other, so is a factor common to both terms. The variable appears as in one term and as in the other, so is a factor common to both terms. The variable appears as in both terms, so is a factor common to both terms. Taking out common factors, and , so
 Polynomials-and-rational-expressions/706842: Please put the following polynomial into a simpler form involving a fraction: . I need it to compute the area of the polygon whose vertices are the solutions in the complex plane to this same polynomial.1 solutions Answer 435702 by KMST(1863)   on 2013-01-28 20:34:43 (Show Source): You can put this solution on YOUR website! is the sum of a geometric sequence. I always rediscover the formula for a sum, because it's just as easy as looking it up a cumbersome formula and applying it. --> -->
 Geometry_Word_Problems/706747: You can make an electromagnet by wrapping a wire around a nail and then sending a current through the wire.The strength of the magnet is directly proportional to the number of wrappings. The nail, (pictured in Homework Paper, has a diameter of 0.4 Cm). The wire is 0.05 cm in diameter, and you can get a 100 wrappings side-by-side on the nail before having to start another layer. Consider each wrapping a perfect circle. How many meters of wire do you need to the nearest 0.1 m, to make a magnet with 700 wrappings?1 solutions Answer 435701 by KMST(1863)   on 2013-01-28 20:24:19 (Show Source): You can put this solution on YOUR website!Considering each wrapping a perfect circle, the first 100 wrappings would be 100 circles of radius 0.2 cm, diameter 0.4 cm. (I'm counting the radius as going from the very center of the nail to the surface where the nail touches the wire, not the slightly larger diameter to the very center of the wire). That would account for per wrapping or for the first 100 wrappings. The next 100 wrappings are wrapped around nail plus one layer of wire, with a radius of and a radius of The total length of wire for that second layer would be . The third layer would be wrapped around the nail plus two layers of wire with a new, larger diameter of and would total of wire. The length of wire for each layer keeps increasing, and the total length of wire for the 7 layers, meaning 700 wrappings, on diameters measuring, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 cm is That is .
 Triangles/707223: What is the range of lengths of each leg of an isosceles triangle if the measure of the base is 6?1 solutions Answer 435696 by KMST(1863)   on 2013-01-28 20:01:16 (Show Source): You can put this solution on YOUR website!The legs must be longer than 3, but can be as long as you want.
 Polynomials-and-rational-expressions/707045: Can you please explain to me how to solve this equation? I have tried to do it for an hour and cannot get it. 8y^3-27 / 64y^3-1 x 16y^2+4y+1 / 4y^2-9 Im not sure how to make these sit on top of each other on here. But please someone help me!!!!1 solutions Answer 435515 by KMST(1863)   on 2013-01-28 06:24:30 (Show Source): You can put this solution on YOUR website!When I want to make an expressions sit on top of another, I use the special codes from this website. I write the algebraically correct expression ((8y^3-27)/(64y^3-1))*((16y^2+4y+1)/(4y^2-9))= and wrap it between three {'s and 3 }'s. It's much easier to write it with just paper and pencil, but when talking to or through a machine, you have to use precise language. Your problem must be simplifying that rational expression and that requires factoring. At this point you simplify by dividing both numerator and denominator by the same expressions, which we do by crossing out matching top and bottom expressions, saying that "they cancel each other." = NOTES: 1) You have to remember that there are special products that result in a difference of squares, and in difference or sum of cubes. where and can be any mathematical expression. Also, and , where the quadratic polynomials and cannot be factored further. 2) Factoring is a skill essential to math survival, and it requires practice. You thought you were done with polynomials, but factoring keeps coming back to haunt you, all the way through calculus and beyond. I found it the hard way when I was your age. 3) Parentheses are sometimes needed. , but if you key into your calculator 10-1/9-1= , your calculator, who knows and obeys the rules of order of operations, correctly interprets it as =8.88888...
 Linear_Equations_And_Systems_Word_Problems/707001: What is an inequality equation that covers the entire number line? 1 solutions Answer 435512 by KMST(1863)   on 2013-01-28 05:45:19 (Show Source): You can put this solution on YOUR website! is the simplest example I can think of. works too. That kind of function/inequality is what is probably expected. We could get fancy, as in , but there's no need for that.
 Circles/706690: the circle passes through the points A(-5,2) and B(-1,4) and tangent to the line x-5y=10. 1. radius of the circle 2. equation of a circle in a center-radius form 3. equation of circle in general form1 solutions Answer 435477 by KMST(1863)   on 2013-01-27 23:27:22 (Show Source): You can put this solution on YOUR website!I have a solution (two solutions actually), but there has to be a better way to get to the solution, and the results are so complicated that I suspect a typo in the problem. The midpoint of AB is (-3,3). The slope of the line through A and B is The perpendicular bisector of AB is --> --> The center of a circle passing through A and B has to be on that line, so its coordinates would be (h,k) with The equation of the circle is where is the radius. I can even substitute and get as equation for our circle Substituting the coordinates of B, I get an expression for --> --> --> Now I can write the equation for our circle as Simple. I just have to find . I know the circle is tangent to the line --> so that line and the circle have just 1 intersection point. Substituting into the equation for the circle, I can find that intersection point and more. If that quadratic equation must have just one solution, the discriminant must be zero, so The best I can do with that unwieldy equation is divide everything by 4 to get Applying the quadratic formula That can be simplified to Substituting into , we get with and with The approximate values give us points (-1.95,0.91) and (-72.49,141.98) for centers. Exact values for are and and approximate values are and corresponding to the centers above, respectively. and The equation of the circles are for the small circle, and for the large circle. Asking for the equation of circle in general form is cruel.
 Quadratic-relations-and-conic-sections/706935: how do i factor 3c^2-5c=01 solutions Answer 435438 by KMST(1863)   on 2013-01-27 20:23:22 (Show Source): You can put this solution on YOUR website!Take out as a common factor: --> Either or -->
 Complex_Numbers/706890: A cross section of a hollow circular pipe shows that its inner diameter is 2 1/10 inches and that the pipe is 3/5 inch thick. what is it's outside diameter (OD)? A costume designer bought 2 1/2 yards of black silk, 3yards 6 inches of white silk,and 6 1/3 yards of blue silk. How many yards of silk did she buy? in a track meet four athletes scored 1 1/2, 8 3/4 , 5 1/8, and 1 1/4 points for their team. what is the total number of points they scored? A piece of metal 3 7/8 inches long is to be cut from a longer piece of 5 1/16 inches. what will be the length of the remaining piece? add 7 5/8 and 5 5/8. then from their sum subtract 11 5/16. 15* 4 2/3 Evaluate.1 solutions Answer 435434 by KMST(1863)   on 2013-01-27 19:55:46 (Show Source): You can put this solution on YOUR website!If the thickness of the pipe wall is inch=inch, we have to add to the inner diameter twice that thickness to get the outside diameter, which is inch=inch=inch For the silk problem, we need to get all those numbers as fractions of a yard, rather than mixed numbers with mixed units. 6 inches = (1/2) foot and 1 foot = 1/3 yard, so 6 inches = (1/2) of (1/3) of a yard or of a yard 3yards 6 inches = 3 yards + 1/6 yard = Let's add all those yards of fabric: Now we need a common denominator, and is the least common denominator we could use It was a total of 12 yards. Now we need a common denominator, and is the least common denominator we could use I get by dividing 133 by 8. The quotient is 16 whole units and the remainder, , divided by is . A piece of metal 3 7/8 inches long is to be cut from a longer piece of 5 1/16 inches. what will be the length of the remaining piece? add 7 5/8 and 5 5/8. then from their sum subtract 11 5/16. Now we need a common denominator, and is the least common denominator we could use 15* 4 2/3 Evaluate.
 Surface-area/706825: I posted a problem several days ago, hoping to get some assistance for assignment due on Monday. Need help please. A window in the shape of a rectangle with a semicircular top, (Norman window), . The height of the rectangular part is twice the base. If the base is 6 feet, what is the area? I got 86.13 feet squared. Then: Assume that the same window has a radius of 2 feet.We want the rectangular part of window to have same area as the semicircular top.What should the dimensions of the rectangle be? Please explain.1 solutions Answer 435425 by KMST(1863)   on 2013-01-27 18:55:40 (Show Source): You can put this solution on YOUR website!The area will be the area of the rectangular part, plus the area of the semicircular part. For the second part: If the radius is ft, the diameter, which is also the width of the rectangle, is ft. That would be the base of the downwind. and if the height of the rectangular part is twice the base, then that height is feet=feet. Then the area of the rectangular part is The area of half of a circle with radius is The total area is = approximately . For the first part, a rectangle 3 feet by 6 feet would have an area of 72 square feet, and a semicircle with a 6 feet diameter (3 feet radius) would have an area of =approximately , for a total area of approximately . That is what you found, aside from the rounding difference. (I used a more accurate version of , but I would get 86.13 too, If I used 3.14 por ).