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# Recent problems solved by 'KMST'

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 Geometry_Word_Problems/752371: Ok so here's the question: i have to consider the region R = { (x,y) : x is greater than or equal to 0 and less than or equal to 6 AND y is less than or equal to (2/3)x and greater than or equal to 0 }. Which i know is a triangle. Then i have to sketch the solid of revolution S obtained by revolving R about the x-axis. Which i did. Now here's where i'm stuck. I have to find the formula for the area of the cross-section perpendicular to the x-axis at x=a, which i know will be a circle. So i started out with the formula for area of a circle which is A= pi(r)^2. and i think that i somehow have to use (2/3)x and x=a to solve for r, and then substitute for the "r" in the equation for area so that all i have to do when given an x value later is plug it in for a, but i'm lost as to how to do that...any idea?1 solutions Answer 457730 by KMST(1874)   on 2013-05-25 08:15:56 (Show Source): You can put this solution on YOUR website!Your idea is good, but to convince yourself, you need a drawing or two (or a very good imagination). Here are my drawings. Let's loox at the x-y plane, and consider points in region R that at part of that cross section at . They form segment AP, that goes from the x-axis to that green line. Points A and P have . The radius of the circle is AP Since P is on the line, its y-coordinate is , and that is the distance AP, the radius of the circle. So --> or
 Angles/752391: if two supplementary angles differ by 44 degree ,then one of the angle is?1 solutions Answer 457729 by KMST(1874)   on 2013-05-25 06:40:25 (Show Source): You can put this solution on YOUR website!If two supplementary angles differ by , thein their measures, in degrees are and Since they are supplementary their measures add to , so From that equation, we can find --> --> --> --> --> -->
 test/752418: Wat irrational numba is betwin 2 & 3.1 solutions Answer 457728 by KMST(1874)   on 2013-05-25 06:30:21 (Show Source): You can put this solution on YOUR website!Between 2 and 3 there are infinite irrational numbers (and infinite rational numbers too). ONE IDEA: If is between 2 and 3, then means that , which means So is a solution, and so are , , and . FANCIER IDEAS: You probably know that is an irrational number that is between 3 and 4. So means that . is another irrational number between 2 and 3. FANCIER: You may not know it, but there is an irrational number called that is approximately 2.7183. It is sort of like in that there is no other way to express it, and to describe it you have to tell a story. However, the story of is more complicated than the story of .
 Polynomials-and-rational-expressions/752297: A polynomial functions f has x- intercepts at -5,-3,0,6? Which is a possible equation for the function? A. f(x)=(x+5)(x+3)(x-6) B. f(x)=x(x+5)(x+3)(x-6) C. f(x)=(x-5)(x-3)(x+6) D. f(x)=x(x-5)(x-3)(x+6) I typed it up exactly how it is on the review. (This is not a question on a test, this is a question on the review for the test. SO I really need to know how to solve this)1 solutions Answer 457707 by KMST(1874)   on 2013-05-24 22:20:39 (Show Source): You can put this solution on YOUR website!If the function has x- intercepts at -5, -3, 0, and 6, it means that f(x) is zero for those values of x. As a consequence, when the function is factored, it must have all of the following factors: (x-(-5))=(x+5), which becomes zero when x=-5, (x-(-3))=(x+3), which becomes zero when x=-3, (x-0)=x, which becomes zero when x=0, and (x-6), which becomes zero when x=6. (It could have other factors too, but it must have those four factors). So . is the only option that works.
 Finance/752253: Good Morning, I have a mark-up problem that is haunting me. Could you please show me how to set it up correctly? I can't seem to figure it out. Here it is: A pair of shoes costs the retailer \$85 per pair. At what price should the retailer mark them so he can sell them at a 15% disoount off the selling price and still make a 20% profit on his cost? My brain is frozen on this. Thank you in advance for your help. Kindest regards, Michelle Dean1 solutions Answer 457650 by KMST(1874)   on 2013-05-24 12:43:02 (Show Source): You can put this solution on YOUR website!= marked "original" price (in \$) discounted price = The profit would be the discounted price minus the cost, profit = As a percentage of the \$85 cost, the profit would be x100%=20% or --> --> --> --> --> --> The retailer should mark the original price as \$120. The customer would get a 15% discount, which would reduce the price by \$18, to \$102. Selling the shoes at \$102, the retailer would have a profit of \$102 - \$85 = \$17, and that \$17 is 20% of the \$85 cost.
 Quadratic-relations-and-conic-sections/750433: i dont understand how to graph the vertical ellipse whose center is at (3,2), minor axis is 6, and has a vertex at (3,-3). please help 1 solutions Answer 456581 by KMST(1874)   on 2013-05-19 15:51:43 (Show Source): You can put this solution on YOUR website!You are given the coordinates of the center and one vertex, so you can plot those point right away. You are also given the horizontal width of ellipse, the 6 units that are the length of the minor axis. I indicated the above information in red on the diagram below: The vertex given is 5 units below the center, so the other vertex must be 5 units above the center, at (3,7). Since the whole minor axis is 6 units long, each end will be 3 units from the center. That puts the ends of the minor axis at (0,2) and (6,2). Those two points are called the co-vertices. The second vertex and the co-vertices are marked in blue in the diagram. The next and final step is sketching a curve that looks like an ellipse and goes through the vertices and co-vertices. The ellipse is drawn in green in the diagram. If you sketch your ellipse by hand, going through the vertices and co-vertices it should be acceptable, even though the other points will not be accurately placed. NOTE: If it was required, you would have to locate the foci. The distances from the center to the vertices, co-vertices, and foci are represented by a, b, and c. = distance to vertex (semi-major axis) = distance to co-vertex (semi-minor axis) = distance to focus (focal distance) They are related by In your case, --> --> --> So the foci wouldbe 4 units above and below the center, at (3,6) and (3,-2).
 Polynomials-and-rational-expressions/750401: LCM I couldn't figure it out, here is work problems: x^2+4x, x^2 -4 and find the LCM 1 solutions Answer 456573 by KMST(1874)   on 2013-05-19 15:04:55 (Show Source):
 Triangles/750316: show that the points (a,0),(0,b) and 3a,-2b) lie on a straight line. find its equation1 solutions Answer 456547 by KMST(1874)   on 2013-05-19 12:04:09 (Show Source): You can put this solution on YOUR website!Two points determine a line. We can calculate the slope of the line between 2 of those points. From the slope and the coordinates of one of the points, we can get the equation of the line. If the coordinates of the third point satisfy that equation, the third point lies on that same line. The slope of the line connecting (a,0) to (0,b) is Since the y-intercept is at (0,b), b is the intercept, and we can write as the slope-intercept form of the equation of the line connecting (a,0) to (0,b). Substituting the x-coordinate of (3a,-2b) into the equation we can find if that point lies on the same line. For , the point on the line has so point (3a,-2b) lies on the same line as the other two points. If we were not asked for the equation of the line, we could calculate the slopes for two different pairs of points. If we found the same slopes connecting two of the points with the other point, that would mean they all lie on the same line.
 Points-lines-and-rays/750024: Rectangle QRST between curve and . Let P be the point of intersection of the side QT and the x-axis. Let α be the length of the perimeter of this rectangle. We are to find the x-coordinate of the point P where α is maximized and also to find the maximum value of α. P(x,0), where 0 Therefore, when x=(A), α is maximized and its maximum value is (B) solve for A and B ((this is the picture of the graph http://i44.tinypic.com/30sk9ir.jpg)1 solutions Answer 456479 by KMST(1874)   on 2013-05-18 22:36:36 (Show Source): You can put this solution on YOUR website!Both curves and the rectangle QRST are symmetrical with respect to the y-axis. TI'll call the coordinates of P (a,0), to distinguish that x-coordinate value from the variable . The x-coordinates of points R and S are the same . The y-coordinate of points Q and R, on curve is . The y-coordinate of points S and T, on curve is . The width ST (or QR) of the rectangle is . The height RS (or QT) of the rectangle is . The perimeter of the rectangle is is a quadratic function in It's maximum is at because a parabola such as has a vertex as The equation in vertex form would be So the maximum value of happens at and the maximum value of is
 Polynomials-and-rational-expressions/750022: consider the integral expression in x! where a is rational number at a=[A], the value of P is a rational number for any x which satisfies the equation ,and this case the value of P is [B} solve for A and B please, im stuck here 1 solutions Answer 456450 by KMST(1874)   on 2013-05-18 18:33:41 (Show Source): You can put this solution on YOUR website! Dividing by we get for quotient with for a remainder. So For any x which satisfies the equation , so
 Graphs/750103: x^3 ______ +1= 10 4 ______ x Can be written as x^4+ax+b=0 Find the values of a and b I would once again like to request someone's help, and if the person can show the working of the solution I would really be grateful for it.1 solutions Answer 456447 by KMST(1874)   on 2013-05-18 18:10:49 (Show Source): You can put this solution on YOUR website! can be written as (x^3+1)/10=4/x Multiplying both sides of the equal sign times we get the equivalent equation --> --> Comparing it to we find and
 Linear-equations/750230: I need help figuring out a problem please. Point A is the origin, find the coordinates of the two points that are 5 units distant from A, and have an x-coordinate of +3. (The two lines should have coordinates of (3,y)). help please! thank you!1 solutions Answer 456445 by KMST(1874)   on 2013-05-18 18:04:19 (Show Source): You can put this solution on YOUR website!The 2 points have y=4 and y=-4. They are on the line . Their coordinates are (3,-4) and (3,4). Each one forms a right triangle with other vertices A(0,0) and (3,0) The legs of those right triangles have lengths and , and the hypotenuse has length
 Radicals/750003: Cube root 40x^4y^71 solutions Answer 456431 by KMST(1874)   on 2013-05-18 16:53:46 (Show Source):
 Radicals/750000: 5th root 64x^8y^21 solutions Answer 456428 by KMST(1874)   on 2013-05-18 16:49:29 (Show Source):
 Radicals/749998: 4th root of 32y^21 solutions Answer 456426 by KMST(1874)   on 2013-05-18 16:43:11 (Show Source):
 Quadratic_Equations/749493: let α be a real number, let us translate the graph of the cubic function .....{1) so that the point (α,f(α)) on the graph (1) is translated into the origin (0,0), and express the function of the translated graph in terms of f'(α) and f"(α) next we consder the translation which translates the point (α,f(α)) on the graph of (1) into the origin, we replace x with x+α and y with y+f(α) in (1), and obtain the expression y=x^3+ f"(α).x^2/A + f'(α)x As an example, consider the function ....(2) f'(4)=0 and f"(4)=0 we see that when we translate the graph of (2) so that the point (B,C) on the graph is moved to the origin, we get the graph of solve for [A] [B] and [C] 1 solutions Answer 456422 by KMST(1874)   on 2013-05-18 15:39:31 (Show Source): You can put this solution on YOUR website!This problem was posted as problem # 749493 (2013-05-16 12:20:44) and as problem # 750070 (2013-05-18 05:10:05). Each time something is was being lost in translation, but it helped to be able to listen to the message twice. One way to translate a graph so that point (a, f(a)) moves to the origin, point (0, 0), is to replace with and with and then solve for When we do that to we get The first and second derivatives of are and so and Comparing to we see that the coefficient of is indeed and is the coefficient of so and How would I use all of the above to find the coordinates of the point (B, C) in the graph of that when translated to the origin turns the function into ? I wouldn't. I would realize that and that which is translated 4 units to the left and 4 units down, and that is the translation that would bring point (B, C) = (4, 4) to (0, 0). That looks to me like the most efficient way to the solution. Or maybe after being told that translated turns into and that I would realize that must have just one inflection point, just like . Since I know that has its inflection point at (0, 0), I would realize that the inflection point of at must be the point translated to the origin. Then I would know that and would only need to calculate the y-coordinate of the inflection point, --> --> --> --> But maybe we are supposed to use the first part and realize that with it would man that translating (4, f(4)) into the origin would transform into and if and the equation transforms into
 Graphs/750070: let a be a real number, let us translate the graph of the cubic function .....{1) so that the point (a,f(a)) on the graph (1) is translated into the origin (0,0), and express the function of the translated graph in terms of f'(a) and f"(a) next we consder the translation which translates the point (a,f(a)) on the graph of (1) into the origin, we replace x with x+a and y with y+f(a) in (1), and obtain the expression f"+f' As an example, consider the function ....(2) f'(4)=0 and f"(4)=0 we see that when we translate the graph of (2) so that the point (B,C) on the graph is moved to the origin, we get the graph of solve for [A] [B] and [C] 1 solutions Answer 456421 by KMST(1874)   on 2013-05-18 15:39:00 (Show Source): You can put this solution on YOUR website!This problem was posted as problem # 749493 (2013-05-16 12:20:44) and as problem # 750070 (2013-05-18 05:10:05). Each time something is was being lost in translation, but it helped to be able to listen to the message twice. One way to translate a graph so that point (a, f(a)) moves to the origin, point (0, 0), is to replace with and with and then solve for When we do that to we get The first and second derivatives of are and so and Comparing to we see that the coefficient of is indeed and is the coefficient of so and How would I use all of the above to find the coordinates of the point (B, C) in the graph of that when translated to the origin turns the function into ? I wouldn't. I would realize that and that which is translated 4 units to the left and 4 units down, and that is the translation that would bring point (B, C) = (4, 4) to (0, 0). That looks to me like the most efficient way to the solution. Or maybe after being told that translated turns into and that I would realize that must have just one inflection point, just like . Since I know that has its inflection point at (0, 0), I would realize that the inflection point of at must be the point translated to the origin. Then I would know that and would only need to calculate the y-coordinate of the inflection point, --> --> --> --> But maybe we are supposed to use the first part and realize that with it would man that translating (4, f(4)) into the origin would transform into and if and the equation transforms into
 Polynomials-and-rational-expressions/750122: what is 2y^3-12y^2+18y1 solutions Answer 456400 by KMST(1874)   on 2013-05-18 13:23:27 (Show Source):
 Square-cubic-other-roots/750116: what is the square root of 841 i am stuck i am not getting the answer i am getting 42 an if you got the answer pls tell me th method 1 solutions Answer 456364 by KMST(1874)   on 2013-05-18 11:25:00 (Show Source): You can put this solution on YOUR website! is close to and a little less than makes so and NOTE: For other numbers, you could get at the answer by doing a prime factorization.
 Miscellaneous_Word_Problems/750071: If 8 students are seated on each bench ,3 benches are left over .If 5 students are seated on each bench,12 students are left over How many benches students are there?1 solutions Answer 456337 by KMST(1874)   on 2013-05-18 08:42:57 (Show Source): You can put this solution on YOUR website!AS A SYSTEM OF EQUATIONS PROBLEM: = number of students = number of benches "If 5 students are seated on each bench, 12 students are left over" translates as "If 8 students are seated on each bench, 3 benches are left over" cound be translated as if we are creative enough to think that the 3 empty benches could have housed extra students. Otherwise, we could translate "If 8 students are seated on each bench, 3 benches are left over" literally as since is the number of benches that are occupied. Of course, the two equations are equivalent: --> --> --> --> --> --> --> --> --> AS A GUESS-AND-CHECK/DIVISIBILITY PROBLEM: Since all students fit into a number of benches with students on each bench, the number of students is a multiple of . Sitting students per bench you fill all the benches and have 12 students left over. You could start guessing and checking from there, but it may be time consuming. You could have 4 or more benches, and 8, 16, 24, 32, 40, ... students. You can try each number, or skip some. Making a table can help. It would be better to reduce the number of choices to try. Guessing away without further thought: You could imagine having 4 benches. With 8 students, you can fill 1 bench with 8 students and would have 3 benches left. But with a total of 4 benches, sitting 5 on each bench you need students to fill the benches and have 12 students left over. It must be more than 8 students. With 5 benches, you can fill 2 benches with 8 students each (a total of 16 students), and would have 3 benches left. But 5 benches at 5 students per bench would sit 25 students, so you could not fill those 5 benches and have 12 students left over, so there are more than 5 benches and more than 16 students. With 6 benches, you could fill 3 benches with 8 students each (24 student total), and would have 3 benches left. But 6 benches at 5 students per bench would sit 30 students, so 6 benches/24 students is still too low. Let's skip 7 benches and go to 8 benches. With 8 benches, you could fill 5 benches with 8 students each (40 students), and would have 3 benches left. But 8 benches at 5 students per bench would sit 40 students, with no student left over, so the numbers of benches/students at 8/40 are still too low. Let's skip further and go to 13 benches. With 13 benches/80 students, you could fill 10 benches with 8 students each and would have 3 benches left. But 31 benches at 5 students per bench would sit 65 students, with students left over, and that is too many. The numbers of benches/students are less than 13/80, so we may try the next smaller guess, 12 benches/72 students. With 12 benches/72 students, you could fill 9 benches with 8 students each and would have 3 benches left. With 12 benches, at 5 students per bench, you would sit 60 students, with students left over, and that is exactly what the problem asks for. So benches and students is the answer. ALTERNATIVELY, you can use the information to reduce the number of guesses to try to 4 benches/32 students, 12 benches/72 students, 4 benches/32 students, 20 benches/112 students, etc. One way: Sitting students per bench you fill all the benches and have 12 students left.T hat means you could fill more benches with another students and have just students without a seat. That tells you that the number of students is more than a multiple of . The multiples of 5 that are not odd rather than even (like 5,15, 15, 35, etc) would not work, because adding to those numbers will still give you and odd number for the total number of students, and the total number of students is a multiple of 8, so it must be even. Now we know that the total number of students should be an even multiple of 5 (a multiple of 10) plus 2, and it must also be a multiple of 8. The total number of students cannot be 12, or 22, or 42, or 52, or 62, or 82, or 92, or 102, or 122 (not multiples of 8), but we should try 32, 72, and 112. A craftier way (maybe too creative) to get at the same point: If I gave you 8 more students, and 4 more benches you could have everyone comfortably seated at 5 students per bench. With the 8 added students, the number of students would be a multiple of 8 and a multiple of 5, so it would be a multiple of 40. The number of benches would be a multiple of 8 and the number of students would be a multiple of 40, as in 8/40, 16/80, 24,120. That means that before being given the 4 extra benches/8 extra students you had benches/ students, or benches/ students, or benches/ students, or ... At 8 students per bench, 32 students use benches, using all 4 benches with no bench left over (too few); and 112 students use benches, with benches left over (too many); and 72 students use benches, with benches left over (just right).
 Travel_Word_Problems/749949: if a car takes 2 hours to travel 100km what is the cars average speed?1 solutions Answer 456263 by KMST(1874)   on 2013-05-17 21:09:01 (Show Source):
 t e s t/749851: Evaluate a) (32)^(0.6) - (1/16)^(0.75) Can you please help me out? Thanks so much in advance Can you also please show the steps it would really help me understand:)1 solutions Answer 456184 by KMST(1874)   on 2013-05-17 12:44:04 (Show Source): You can put this solution on YOUR website! and so and so Putting it all together:
 logarithm/749489: [log(x-4)=1-log(x-1)] + [the distance between (2,-3) and (-8, -3+ square root of 21)] I tried solving the distance part and got 14.6, but I think its wrong. I have no idea where to begin with the log part.1 solutions Answer 456155 by KMST(1874)   on 2013-05-17 07:50:22 (Show Source): You can put this solution on YOUR website!This diagram below shows points (2,-3) and (-8, ) circled, and the distance between them The distance between points (2,-3) and (-8, can be calculated as where is the difference between the x-coordinates of the points, and is the difference between the y-coordinates of the points. It is not a difficult calculation, and no complicated distance formula needs to be memorized. (If your teacher disagrees, and requires that you It is not a difficult calculation, and no complicated distance formula needs to be memorized. You do not even have to worry about the absolute values or the order of the numbers you subtract to get those differences because after you square them, it does not matter if it was and ; you get the same squared difference. NOTE: If your teacher disagrees, and requires that you make it look complicated, you may have to write something like After finding that distance, the problem log(x-4)=1-log(x-1) + [the distance between (2,-3) and (-8, -3+ square root of 21)] turns into --> --> --> --> The equation --> --> makes me suspect some typo in the problem. IF it had been log(x-4)=-10-log(x-1) + [the distance between (2,-3) and (-8, -3+ square root of 21)] , it would simplify to --> --> --> --> --> --> with solutions and , and verifying in the original equation we would see that is a solution of , but does not work because it makes and and their logarithms would not exist.
 logarithm/749494: consider the curve y=2logx, where log is the natural logarithm. let α be the tangent to that curve which passes through the origin, let P be the point of contact of α and that curve, and let m be the straight line perpendicular to the tangent α at P. We are to find the equations of the straight lines α and m and the area S of the region bounded by the curve y=2logx, the straight line m, and the x-axis let t be the x-coordinate of the poin P, then t satisfies log t=(A). Hence the equation of α is the equation of m is thus the area S of the region is solve for A,B,C,D,E,F and G1 solutions Answer 456151 by KMST(1874)   on 2013-05-17 06:46:11 (Show Source): You can put this solution on YOUR website!The function is Its graph crosses the x-axis at the point where --> --> --> The x-coordinate of point P is . The slope of the tangent at point P is the value of the derivative at that point. y'=, so the slope of the tangent at is . Since the line tangent at P passes through the origin, its equation must be At point P, with , --> Since point P is on the graph of , its y-coordinate is So --> --> --> and P is (e,2). Now we can find the equation of : --> --> is , line perpendicular to must have a slope of . As passes through P(e,2) its equation is --> --> --> So , , and The line crosses the x-axis at the point where --> --> --> --> --> The area of the region bounded by the curve , the straight line , and the x-axis is shown below. can be can be calculated as the sum of: the area below , and above the x-axis, between and , plus the area below between and , is easier than it seems. It's just the area of the triangle with vertices (e,0), P(e,2), and (e+e/4,0) Its base is ; its height is , and its area is . Since , So and
 Exponential-and-logarithmic-functions/749560: I can not seem to figure this question out... log[4](x+3)-log[4](x+2) >= 3/21 solutions Answer 456122 by KMST(1874)   on 2013-05-16 22:48:35 (Show Source): You can put this solution on YOUR website!For and to exist, it must be that <--> (because logarithm exist only for positive numbers). --> --> --> --> --> --> Since , multiplying both sides times does not require flipping the inequality sign, so --> --> --> --> --> The solution is
 Inequalities/749565: (-3,-3) (3,-1)Its a dotted line and the shaded part is on tom how do i write this slope-intercept inequality 1 solutions Answer 456117 by KMST(1874)   on 2013-05-16 21:43:06 (Show Source): You can put this solution on YOUR website!A dotted line going through (-3,3) and (3,1) is not part of the solution (if it were it would be drawn as a solid line to indicate that), but it is a boundary. The slope of that line is The equation can be written in point=slope form based on point (3,-1) as --> --> --> The slope-intercept form of the line (the form that starts with y=...) is The area shaded above that line represents