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# Recent problems solved by 'lwsshak3'

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 Length-and-distance/593538: a triangle has vertex coordinates (1,1), (1,5) and (5,3). find the length of each side and decide what type of triangle it is.1 solutions Answer 376353 by lwsshak3(6761)   on 2012-03-30 02:02:04 (Show Source): You can put this solution on YOUR website!a triangle has vertex coordinates (1,1), (1,5) and (5,3). find the length of each side and decide what type of triangle it is. Using distance formula: d^2=(x1-x2)^2+(y1-y2)^2 .. Between (1,1) and (1,5) d^2=(1-1)^2+(5-1)^2=0+16=16 d=4 .. Between (1,5) and (5,3) d^2=(5-1)^2+(3-5)^2=16+4=20 d=√20 .. Between (5,3) and (1,1) d^2=(5-1)^2+(3-1)^2=16+4=20 d=√20 .. This is an isosceles triangle with two of the sides=√20 and the third side=4
 Trigonometry-basics/593464: 3cos^2a+2=3-2cosa1 solutions Answer 376330 by lwsshak3(6761)   on 2012-03-29 21:30:26 (Show Source): You can put this solution on YOUR website!3cos^2a+2=3-2cosa 3cos^2a+2cosa-1=0 (3cosa-1)(cosa+1)=0 .. 3cosa-1=0 3cosa=1 cosa=1/3 a≈70.5º or cosa+1=0 cosa=-1 a=180º
 Geometry_Word_Problems/593448: The endpoint coordinates of the hypotnuse of a right triangle are (1,0) and (6,-5). What are possible coordinates of the point where the legs of this triangle intersect and what is the lenght of the hypotenuse? 1 solutions Answer 376323 by lwsshak3(6761)   on 2012-03-29 21:11:55 (Show Source): You can put this solution on YOUR website!The endpoint coordinates of the hypotnuse of a right triangle are (1,0) and (6,-5). What are possible coordinates of the point where the legs of this triangle intersect and what is the lenght of the hypotenuse? ** The best way to see this problem is to roughly plot the points. You will see that the two legs of the right triangle intersect at (0,6). This makes the vertical leg=5 and the horizontal leg=6. The length of the hypotenuse by Pythagorean Theorem=√(6^2+5^2)=√(36+25)=√61
 Inequalities/593450: Solve the linear inequality and write the solution in set-builder notation. 8x-38˂101 solutions Answer 376319 by lwsshak3(6761)   on 2012-03-29 20:55:07 (Show Source): You can put this solution on YOUR website!Solve the linear inequality and write the solution in set-builder notation. 8x-38˂10 ** 8x-38<10 8x<48 x<6 {x|x<6}
 Money_Word_Problems/593443: How long will it take \$750 to quadruple in an account that pays 4.5% interest compounded continuously?1 solutions Answer 376315 by lwsshak3(6761)   on 2012-03-29 20:50:09 (Show Source): You can put this solution on YOUR website!How long will it take \$750 to quadruple in an account that pays 4.5% interest compounded continuously? ** Formula for continuous compounding:A= Pe^rt, P=initial investment, r=interest per period, t=number of periods, A=amount after t periods. For given problem: P=750 r=4.5% per year t=number of years A=4P .. A/P=e^rt 4=e^rt=e^.045t take log of both sides .045t*lne=ln4 log of base=lne=1 .045t=ln4 t=ln4/.045 t=30.8 ans: It would take about 40 years to quadruple in an account that pays 4.5% interest compounded continuously? note: this would be true for any initial investment compounded continuously at 4.5% interest rate.
 Quadratic-relations-and-conic-sections/593014: What is the focus, vertex, axis of symmetry and directrix of the parabola equation y2-8y+16x-64=01 solutions Answer 376191 by lwsshak3(6761)   on 2012-03-29 03:21:29 (Show Source): You can put this solution on YOUR website!What is the focus, vertex, axis of symmetry and directrix of the parabola equation y2-8y+16x-64=0 complete the square (y^2-8y+16)+16x-64-16=0 (y-4)^2+16x-80=0 (y-4)^2=-16x+80 (y-4)^2=-16(x-5) This is an equation for a parabola that open leftwards of the standard form: (y-k)^2=-4p(x-h), (h,k)=(x,y) coordinates of the vertex. For given parabola: 4p=16 p=4 vertex: (5,4) axis of symmetry: y=4 focus: (1,4) (p units to left of vertex on axis of symmetry) directrix: x=9 (p units to right of vertex on axis of symmetry)
 Quadratic-relations-and-conic-sections/593015: Find an equation for the parabola with focus (4,-6) and vertex at (3, -6)1 solutions Answer 376190 by lwsshak3(6761)   on 2012-03-29 02:58:57 (Show Source): You can put this solution on YOUR website!Find an equation for the parabola with focus (4,-6) and vertex at (3, -6) ** Standard form of equation for a parabola that opens rightwards: (y-k)^2=4p(x-h), (h,k)=(x,y) coordinates of vertex. For given parabola: vertex: (3,-6) axis of symmetry: y=-6 p=1 (distance from focus to vertex on axis of symmetry) 4p=4 Equation of parabola: (y+6)^2=4(x-3)
 Mixture_Word_Problems/593099: How much Brand A fruit punch (20% fruit juice) must be mixed with 8 L of Brand B fruit punch (50% fruit juice) to create a mixture containing 40% fruit juice? I need a straight answer please.1 solutions Answer 376189 by lwsshak3(6761)   on 2012-03-29 02:36:40 (Show Source): You can put this solution on YOUR website!How much Brand A fruit punch (20% fruit juice) must be mixed with 8 L of Brand B fruit punch (50% fruit juice) to create a mixture containing 40% fruit juice? ** let x=liters of Brand A fruit punch to be mixed with 8 liters of Brand B fruit punch x+8=liters of the mixture .. 20%x+50%*8=40%(x+8) .2x+4=.4x+3.2 .2x=.8 x=4 liters ans: liters of Brand A fruit punch to be mixed with 8 liters of Brand B fruit punch=4
 Equations/593054: How to solve the value of x? 1 solutions Answer 376188 by lwsshak3(6761)   on 2012-03-29 02:17:24 (Show Source): You can put this solution on YOUR website!(7/3)x^(4/3)-3x^(-2/3)= 0 x^(-2/3)((7/3)x^(6/3)-3)=0 x^(-2/3)=0 1/x^(2/3)≠0 no solution or (7/3)x^2-3=0 (7/3)x^2=3 7x^2=9 x^2=9/7 x=±√(9/7)=±3/√7=±3√7/7
 Travel_Word_Problems/593076: A boy walked over hills at 3 kilometers per hour and on flat roads at 4 kilometers per hour. He walked 18 kilometers in 5 hours. How much of his trip was over hills?1 solutions Answer 376181 by lwsshak3(6761)   on 2012-03-29 01:20:12 (Show Source): You can put this solution on YOUR website!A boy walked over hills at 3 kilometers per hour and on flat roads at 4 kilometers per hour. He walked 18 kilometers in 5 hours. How much of his trip was over hills? ** let x=distance walked over hills 18-x=distance walked over flat roads travel time=distance/speed .. x/3+(18-x)/4=5 LCD:12 4x+54-3x=60 x=6 km ans: distance walked over hills=6 km
 Equations/592839: Add 1 solutions Answer 376119 by lwsshak3(6761)   on 2012-03-28 18:17:30 (Show Source): You can put this solution on YOUR website!(4x+12)/(x-2) - (2x+8)/(-x+2) (4x+12)/(x-2) + (2x+8)/(x-2) 4x+12+2x+8/(x-2) (6x+20)/(x-2) 2(3x+10)/(x-2)
 Radicals/592837: simplify 1 solutions Answer 376115 by lwsshak3(6761)   on 2012-03-28 18:07:15 (Show Source): You can put this solution on YOUR website!5/ (3) sqrt(5) + sqrt (3) 5/(3√5+√3) multiply top and bottom by the conjugate of the denominator=(3√5-√3) 5(3√5-√3)/(3√5+√3)(3√5-√3) (15√5-5√3)/(45-3) (15√5-5√3)/(42)
 Radicals/592831: 1 solutions Answer 376111 by lwsshak3(6761)   on 2012-03-28 17:44:56 (Show Source): You can put this solution on YOUR website!4/ (6-2 (sqrt (6) 4/(6-2√6) multiply top and bottom by conjugate of denominator=(6+2√6) 4(6+2√6)/(6-2√6)(6+2√6) (24+8√6)/(36-24)(difference of squares) (24+8√6)/12 24/12+8√6/12 2+2√6/3
 Rational-functions/592852: why isn't (1,3) (2,2) (1,5) (2,6) a function1 solutions Answer 376108 by lwsshak3(6761)   on 2012-03-28 17:25:09 (Show Source): You can put this solution on YOUR website!why isn't (1,3) (2,2) (1,5) (2,6) a function For an expression to be a function, for each x there must only be only one y. You will note that for points (1,3) and (1,5), x has two different values y. Similarly, (2,2 and (2,6) show the same. Therefore, these points are not from a function. If you graph these points, you will get a curve similar to a parabola opening rightwards and it would fail the vertical line test which allows only one intersection for a function.
 Inverses/589966: Decide whether or not the functions are inverses of each other. 1] f(x)= 4x+16 and g(x)= 1/4x-4 2] f(x)= sqrt(x+8), domain [-8, infinity); g(x)=x^2+8, domain (-infinity, infinity) and last one: Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. f(x) = 2x + 1 I need help understanding these three questions :( Inverses was my least favorite in Algebra! If you could please help, would be greatly appreciated!1 solutions Answer 376098 by lwsshak3(6761)   on 2012-03-28 17:07:47 (Show Source): You can put this solution on YOUR website!Decide whether or not the functions are inverses of each other. 1] f(x)= 4x+16 and g(x)= 1/4x-4 2] f(x)= sqrt(x+8), domain [-8, infinity); g(x)=x^2+8, domain (-infinity, infinity) and last one: Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. f(x) = 2x + 1 ** 1] f(x)= 4x+16 and g(x)= 1/4x-4 (fog)(x)=f[g(x)]=4[(x/4)-4)]+16=x-16+16=x (gof)(x)=g[f(x)]=(1/4)(4x+16)-4=x+4-4=x Therefore functions are one-to-one and inverses to each other. .. 2] f(x)= sqrt(x+8), g(x)=x^2+8 f(x) is a one-to-one function but g(x) is not. g(x) is a parabola which is bumped 8 units up and it would fail the horizontal line test as it would have two intersections, that is for a given y, you could have two different x's. These two functions are not inverses of each other. .. f(x)=2x+1 i) domain:all real numbers or (-∞,∞) ii) range: (-∞,∞) (This is a straight line with a slope=2 and y-intercept=1. iii) the domain of the inverse. x=2y+1 2y=x-1 y^-1=(x-1)/2=x/2-1/2 (This is a straight line with slope=1/2 and y-intercept=-1/2) domain: (-∞,∞) iv) range of inverse: (-∞,∞)
 Polynomials-and-rational-expressions/592787: I don't understand how to do problems that include three or four problems together such as x^3+5x^2-16x-80 can you help or explain 1 solutions Answer 376058 by lwsshak3(6761)   on 2012-03-28 15:01:17 (Show Source): You can put this solution on YOUR website!I don't understand how to do problems that include three or four problems together such as x^3+5x^2-16x-80 can you help or explain. ** Usually, but not always, problems like this with 4 terms are set up so you can factor the expression by a method called grouping: x^3+5x^2-16x-80 Group terms to get a common factor =x^2(x+5)-16(x+5) factor out common term (x+5) =(x+5)(x^2-16) expand difference of squares =(x+5)(x+4)(x-4)
 Quadratic-relations-and-conic-sections/592609: write the equation of the parabola in standard form x^2 + 2x + 8y + 25 = 01 solutions Answer 376010 by lwsshak3(6761)   on 2012-03-28 02:06:44 (Show Source): You can put this solution on YOUR website!write the equation of the parabola in standard form x^2 + 2x + 8y + 25 = 0 Standard form of equation for a parabola: y=A(x-h)^2+k, (h,k)=(x,y) coordinates of the vertex. complete the square: -8y=(x^2+2x+1)+25-1 -8y=(x+1)^2+24 divide by -8 y=-(1/8)(x+1)^2-3 This is an equation of a parabola that opens downwards with vertex at (-1,-3)
 Travel_Word_Problems/592619: an airplane flying into a headwind travels the 1800-mile flying distance between two cities in 3 hours and 36 minutes. On the return flight, the distance is traveled in 3 hours. Find the air speed of the plane and the speed of the wind, assuming that both remain constant.1 solutions Answer 376009 by lwsshak3(6761)   on 2012-03-28 01:51:29 (Show Source): You can put this solution on YOUR website!an airplane flying into a headwind travels the 1800-mile flying distance between two cities in 3 hours and 36 minutes. On the return flight, the distance is traveled in 3 hours. Find the air speed of the plane and the speed of the wind, assuming that both remain constant. ** let x=speed of plane let c=speed of wind x-c =net speed into wind x+c=net speed with wind speed*travel time=distance 3 hrs and 36 min=3.6 hrs .. (x-c)*3.6=1800 (x+c)*3=1800 .. 3.6x-3.6c=1800 3.0x+3.0c=1800 .. multiply first equation by 3 and second equation by 3.6 10.8x-10.8c=5400 10.8x+10.8c=6480 add equations to eliminate c 21.6x=11880 x=11880/21=550 mph .. 3c=1800-3x=1800-1650=150 c=150/3=50 mph .. ans: speed of plane=550 mph speed of wind=50 mph
 logarithm/592631: Please help me solve this equation: log^7 100-log^7(y-5)=log^7 101 solutions Answer 376008 by lwsshak3(6761)   on 2012-03-28 01:15:12 (Show Source): You can put this solution on YOUR website!Please help me solve this equation: log^7 100-log^7(y-5)=log^7 10 I will assume you mean base 7 instead of exponent 7: log7(100)-log7(y-5)=log7(10) log7(100)-log7(y-5)-log7(10)=0 log7(100)-(log7(y-5)+log7(10))=0 place under single log log7[(100)/(y-5)(10)]=0 convert to exponential form: base(7) raised to log of number(0)=number(100)/(y-5)(10) 7^0=(100)/(y-5)(10)=1 (y-5)(10)=100 10y-50=100 10y=150 y=15
 �t�e�s�t/592432: Solve the equation. (Do not use mixed numbers in the answer.) ln(5x-6)+ln(x) = ln(e) 1 solutions Answer 375952 by lwsshak3(6761)   on 2012-03-27 19:19:12 (Show Source): You can put this solution on YOUR website!Solve the equation. (Do not use mixed numbers in the answer.) ln(5x-6)+ln(x) = ln(e) ln(5x-6)+ln(x) =1 ln[(5x-6)(x)]=1 e^1=5x^2-6x=e 5x^2-6x-e=0 solve by following quadratic formula: a=5, b=-6, c=-e
 Trigonometry-basics/592372: Write the trigonometric expression as an algebraic expression in u. 15) cos (sin-1 u)1 solutions Answer 375940 by lwsshak3(6761)   on 2012-03-27 17:52:37 (Show Source): You can put this solution on YOUR website!Write the trigonometric expression as an algebraic expression in u. 15) cos (sin-1 u) ** (sin-1 u) is an angle whose sin=u/1 (opposite side/hypotenuse) The adjacent side for this angle would then be=√(1^2-u^2)=√(1-u^2) (by pythagorean theorem) Thus, cos (sin-1 u)=√(1-u^2)/1 =√(1-u^2)
 Rate-of-work-word-problems/592336: A drain can empty 3/4 of a sink in 1 minute and the faucet can fill 5/6 of the sink in 1 minute. If the faucet is on and the drain is open, how long will it be before the sink overflows?1 solutions Answer 375931 by lwsshak3(6761)   on 2012-03-27 17:33:34 (Show Source): You can put this solution on YOUR website!A drain can empty 3/4 of a sink in 1 minute and the faucet can fill 5/6 of the sink in 1 minute. If the faucet is on and the drain is open, how long will it be before the sink overflows? ** let x=minutes sink overflows 5x/6-3x/4=1 LCD:12 10x-9x=12 x=12 ans: The sink will overflow in 12 minutes
 Polynomials-and-rational-expressions/592325: Given the polynomial function f(x),find the rational zeros,then the other zeros(that is,solve the equation f(x)=0),and factor f(x) into linear factors. F(x)=x^4+9x^3+17x^2-9x-181 solutions Answer 375919 by lwsshak3(6761)   on 2012-03-27 17:07:42 (Show Source): You can put this solution on YOUR website!Given the polynomial function f(x),find the rational zeros,then the other zeros(that is,solve the equation f(x)=0),and factor f(x) into linear factors. F(x)=x^4+9x^3+17x^2-9x-18 ** Using Rational Roots Theorem: ....0...|......1......9......17......-9......-18 ....1...|......1.....10.....27......18........0 (1 is a zero) ....2...|......1.....11.....37......65......112 (2 is upper bound) ======================= ....0...|......1.....10.....27......18 ..-1...|......1.....9.......18......0 (-1 is a zero) F(x)=(x-1)(x+1)(x^2+9x+18) F(x)=(x-1)(x+1)(x+6)(x+3) zeros are: -6, -3, -1, and 1
 logarithm/591420: solve 4+ln(27x) = 25-2lnx1 solutions Answer 375841 by lwsshak3(6761)   on 2012-03-27 04:06:56 (Show Source): You can put this solution on YOUR website!solve 4+ln(27x) = 25-2lnx ln(27x)+2lnx=21 ln[(27x)(x^2)]=21 ln(27x^3)=21 convert to exponential form e^21=27x^3 take cube root of both sides 3x=e^7 x=e^7/3
 logarithm/591423: use the change of base formula to approximate log5 (6) to 3 decimal places 1 solutions Answer 375839 by lwsshak3(6761)   on 2012-03-27 03:50:45 (Show Source): You can put this solution on YOUR website!use the change of base formula to approximate log5 (6) to 3 decimal places log5(6)=log(6)/log(5)≈1.113
 logarithm/591916: THe quesion is 3^Log base 3 of 2 and there is an option to there not being an answer.1 solutions Answer 375838 by lwsshak3(6761)   on 2012-03-27 03:47:00 (Show Source): You can put this solution on YOUR website!THe quesion is 3^Log base 3 of 2 and there is an option to there not being an answer. 3^log3(2) by definition: base raised to log of number=number (exponential form) so 3^log3(2)=2
 logarithm/592037: log5(1.25*4) 1 solutions Answer 375837 by lwsshak3(6761)   on 2012-03-27 03:40:48 (Show Source): You can put this solution on YOUR website!log5(1.25*4) log5(5)=1
 logarithm/592064: What is the solution(s) to log base 4 of 4x + log base 4 of (x+3) = 2 1 solutions Answer 375836 by lwsshak3(6761)   on 2012-03-27 03:36:41 (Show Source): You can put this solution on YOUR website!What is the solution(s) to log base 4 of 4x + log base 4 of (x+3) = 2 log4[(4x)(x+3)]=2 convert to exponential form: base(4) raised to log of number(2)=number(4x)(x+3) 4^2=(4x)(x+3)=16 4x^2+12x=16 4x^2+12x-16=0 x^2+3x-4=0 (x+4)(x-1)-0 x=-4 (reject,x>0) or x=1
 Quadratic-relations-and-conic-sections/591303: Write an equation in standard form for the circle that statisfies these conditions center(0,7) tangent to x-axis1 solutions Answer 375835 by lwsshak3(6761)   on 2012-03-27 03:12:24 (Show Source): You can put this solution on YOUR website!Write an equation in standard form for the circle that statisfies these conditions center(0,7) tangent to x-axis ** If you make a rough sketch of given points, you will see that the radius=7 Standard form of equation for a circle: (x-h)^2+(y-k)^2=r^2, (h,k)=(x,y) coordinates of center, r=radius. Equation of given circle x^2+(y-7)^2=7^2 x^2+(y-7)^2=49
 Quadratic-relations-and-conic-sections/591913: write an equation in vertex form then graph it. Label center,verticies,co-verticies and the foci> x^2+4y^2=161 solutions Answer 375834 by lwsshak3(6761)   on 2012-03-27 03:01:45 (Show Source): You can put this solution on YOUR website!write an equation in vertex form then graph it. Label center,verticies,co-verticies and the foci> x^2+4y^2=16 divide by 16 x^2/16+y^2/4=1 This is an equation of an ellipse with horizontal major axis of the standard form: (x-h)^2/a^2=(y-k)^2/b^2=1,a>b, (h,k)=(x,y) coordinates of center. For given ellipse: center:(0,0) a^2=16 a=√16=4 vertices: (0±a,0)=(0±4,0)=(-4,0) and (4,0) .. b^2=4 b=√4=2 co-vertices: (0,0±b)= (0,0±2)=(0,-2) and (0,2) .. c^2=a^2-b^2=16-4=12 c=√12≈3.46 Foci:(0±c,0)=(0±√12,0)=(-3.46,0) and (3.46,0) see graph below: y=±(4-x^2/4)^.5