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# Recent problems solved by 'mananth'

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 Travel_Word_Problems/736018: I am a librarian. A patron needs a solution to this problem: The speed of a boat in still waters is 10 km/h The boat travels 12 km upstream and 28 km downstream in a total of 4 hours. What is the speed of the stream? Thanks1 solutions Answer 449635 by mananth(12270)   on 2013-04-10 09:51:37 (Show Source): You can put this solution on YOUR website!let speed of current be c kph upstream speed = 10-c upstream distance = 12 down stream speed = 10+c down stream distance = 28 km upstream time + downstream time = 4 t=d/r 12/(10-c) + 28/(10+c)= 4 multiply the equation by (10+c)(10-c) 12(10+c)+28(10-c)=4(10+c)(10-c) 120+12c+280-28c=4(100-c^2) 400-16c=400-4c^2 re arrange 4c^2=16c /4c c=4 current speed = 4 kph
 Quadratic_Equations/736013: x^2+5x+2=01 solutions Answer 449624 by mananth(12270)   on 2013-04-10 08:31:28 (Show Source): You can put this solution on YOUR website!x^2+5x+2=0 we use the formula method compare with ax^2 +bx +c a= 1 , b= 5 , c= 2 b^2-4ac= 25 + -8 b^2-4ac= 17 x1=( -5 + 4.12 )/ 2 x1= -0.44 x2=( -5 -4.12 ) / 2 x2= -4.56 (-0.44,-4.56)
 Word_Problems_With_Coins/736003: Hey my name is savannah and I do not even know how to begin this problem: Paul E. Bank has \$2.25 in quarters and dimes. He has 15 coins. Find the number of quarters and dimes he has.1 solutions Answer 449622 by mananth(12270)   on 2013-04-10 08:28:43 (Show Source): You can put this solution on YOUR website!quarter is 25 cents dime = 10 cents I am sure you know this (Ha,Ha) total coins = 15 if I have x dime then the balance will be 15-x dimes = x quarters = 15-x Value of dimes = 10x Value of quarters = 25(15-x) total amount is \$ 2.25 = 225 cents 10x+25(15-x)=225 10x+375-25x=225 -15x+375=225 -375 -15x=225-375 -15x=-150 /-15 x=10 the number of dimes balance 5 will be quarters CHECK to make sure 10*10 + 25 *5 = 225 There you are cheers!
 Complex_Numbers/735957: can someone help me figure out how I multiply expressions like (7+9i)(5+i)1 solutions Answer 449620 by mananth(12270)   on 2013-04-10 06:27:32 (Show Source): You can put this solution on YOUR website!(7+9i)(5+i) 7(5+i)+9i(5+i) 35+7i+45i+9i^2 (i^2=-1): 9i^2=-9 26+52i
 Linear-systems/735949: I have tried many times i need serious help!this equation isn't working out right i don't know why, help! Solve the linear system by using substitution: 2x+2y=6 and -3x+5y=(-33) 1 solutions Answer 449617 by mananth(12270)   on 2013-04-10 06:12:03 (Show Source): You can put this solution on YOUR website!2x+2y=6 /2 x+y=3 x= 3-y .. . -3x+5y=(-33) -3(3-y)+5y=-33 -9+3y+5y=-33 8y=-24 /8 y=-3 Plug y in x+y=3 we get x=6 (6,-3)
 Trigonometry-basics/735965: cos 4θ + 4 cos 2θ ≡ 8 cos4 θ − 3. 1 solutions Answer 449616 by mananth(12270)   on 2013-04-10 06:05:53 (Show Source): You can put this solution on YOUR website! cos(4θ) = cos(2*(2θ)) = 2cos^2(2θ) - 1 = 2[ 2cos^2(θ) - 1 ]^2 - 1 2 [ 4cos^4(θ) - 4cos^2(θ) + 1 ] - 1 8cos^4(θ) - 8cos^2(θ) + 1.....................(1) ............. 4 cos 2theta 4(2 cos^2-1) 8 cos^2-4................................(2) (1)+(2)= RHS 8cos^4(θ) - 8cos^2(θ) + 1 + 8 cos^2-4= RHS
 Trigonometry-basics/735545: Angles of Depression/Elevation: From the top of a tower 83 feet high, a man observes that the angle of elevation of the top of a certain tree is 13 degrees and the angle of depression of its base is 72 degrees. Find the height of the tree.''.1 solutions Answer 449463 by mananth(12270)   on 2013-04-09 08:16:51 (Show Source): You can put this solution on YOUR website!tower = 83 ft let the tree be x ft taller than the tower Tree height = (83+x) ft Let y be the distance between tree and tower Tan 13 = x/y y=x/ Tan 13.........................(1) Tan 72 = 83/y y= 83/tan 72.........................(2) x/ tan 13 = 83/ tan 72 x= 83 * tan 13 / tan 72 x= 6,23 ft Height of tree = 83+6.23= 89.23 feet
 Exponential-and-logarithmic-functions/735544: solve the equation by expressing each side as a power of the same base and then equating the exponents 3125^x= 1/sqrt 5 1 solutions Answer 449462 by mananth(12270)   on 2013-04-09 08:06:18 (Show Source): You can put this solution on YOUR website! 5x=-1/2 x=-1/10
 Trigonometry-basics/735516: a) prove that (1+tan A + sec A)(1+cot A-cosecA)=2 b) Prove that Sin 20degree.sin40degree.sin 60degree.Sin80degree = 3/16 c) sin 5A+2sin7A+sin9A/Cos3A+2cos5A+Cos7A = Sin2A+Cos2A.Tan5A d) Show that Sin112+Cos12 / Cos12-Sin12 = Cot35 e) Prove that Sin 7x+Sinx / Cos5x-Cos3x = Sin2x-Cos2x.cotx 1 solutions Answer 449461 by mananth(12270)   on 2013-04-09 08:02:22 (Show Source): You can put this solution on YOUR website!a) prove that (1+tan A + sec A)(1+cot A-cosecA)=2 LHS => = 2 LHS = RHS b) Prove that Sin 20degree.sin40degree.sin 60degree.Sin80degree = 3/16 sin(A+B) = sin A cos B + cos A sin B sin(40) as sin(60-20) sin(80) as sin(60+20) sin(20)•sin(60-20)•[√(3)/2]•sin(60+20) - Simplify sin(60-20)•sin(60+20) sin(60-20) = sin60•cos20 - sin20•cos60 sin(60+20)= sin60•cos20 + sin20•cos60 multiply = sin²60•cos²20 - sin²20•cos²60 cos²A = 1 - sin²A sin²60•cos²20 - sin²20•cos²60 = sin²60•(1-sin²20) - sin²20•(1-sin²60) = sin²60- sin²20•sin²60 - sin²20 + sin²20•sin²60 = sin²60 - sin²20 = 3/4 - sin²20 We get = sin20•[√(3)/2]•(3/4 - sin²20) = [√(3)/8]•(3sin20 - 4sin³20) we know that sin(3A) = 3sin(A) - 4sin³(A) (3sin20 - 4sin³20) = sin60 we have = [√(3)/8]•sin60 = [√(3)/8]•[√(3)/2) = 3/16 LHS = RHS
 Graphs/735495: complet each table with 3 ordered pairs y=1/3 x-3 and graph1 solutions Answer 449439 by mananth(12270)   on 2013-04-09 00:07:33 (Show Source): You can put this solution on YOUR website!y=1/3 x-3 x=3,y=-2 (3,-2) x=6,y=-1, (6,-1) x=9,y=0, (9,0)
 Sequences-and-series/735082: The sum of 3 numbers in an A.P is 21 and their product is 280. find the numbers 1 solutions Answer 449281 by mananth(12270)   on 2013-04-08 09:21:55 (Show Source): You can put this solution on YOUR website!let the numbers be a-d, a , a+d sum = 21 a-d+a+a+d=21 3a=21 a=7 (a-d)(a)(a+d)=280 (a^2-d^2)*a=280 (49-d^2)*7=280 49-d^2= 280/7 49-d^2= 40 d^2=49-40 d^2=9 d= +/-3 (a-d),a,(a+d) if d=3 4,7,10 if a=-3 10,7,4 (Academic)
 Circles/735112: the extremities of the diameter of a circle have coordinates(-4,3)and (6,-2). then the length of the intercept which the circle makes on the y-axis is1 solutions Answer 449279 by mananth(12270)   on 2013-04-08 09:11:09 (Show Source): You can put this solution on YOUR website!Step 1: Find the Midpoint (h,k) of AB: h = A1 + B1 2 h = -4 + 6 2 h = 2 2 h = 1 k = A2 + B2 2 k = 3 + -2 2 k = 1 2 k = 0.5 From above, the center of our circle is (h, k) = (1, 0.5) Calculate radius: r = Square Root((y1 - x1)2 + (y2 - x2)2) r = Square Root((6 - -4)2 + (-2 - 3)2) r = ½Square Root((102 + -52)) r = ½√(100 + 25) r = ½√125 r = ½√125 r = ½(11.1803398875) r = 5.5902 Now calculate our circle equation using (h, k), and r Find the equation of the circle with center (h,k) = (1,0.5) and radius r = 5.5902 The standard equation for a circle is (x - h)2 + (y - k)2 = r2 Plugging in our numbers, we get: (x - 1)2 + (y - 0.5)2 = 5.5902^2 (x - 1)2 + (y - 0.5)2 = 31.25033604 plug x= 0 you will get two values of y add up the two intercepts to get the length of the chord
 Money_Word_Problems/735075: Okay, so I am a very slow learner, and I find it hard to learn things myself if I do not have a fully worked example infront of me. We have just moved onto exam prep and I am revising simple interest The question is At what rate must \$24 000 be invested to earn \$180 per month? My working so far consists of 180 = 24000 x R x T The text book has given us T (time)... being 1/2 I know how to solve, I am just very confused at to why the time is 1/2? Thank you so much Kate 1 solutions Answer 449271 by mananth(12270)   on 2013-04-08 06:50:03 (Show Source): You can put this solution on YOUR website!P= Principal r= rate percent per year t= time in years I= interest earned you need 180 per month 180*12= 2160 per year The formula is I = P * R * T 2160 = 24000 *R *1 ( 1 year) =0.09 0.09*100 = 9% You have to invest at 9% CHECK If I invest \$24000 @9% for 1 year 24000 * 0.09 =2160 2160/12= 180 per month
 Trigonometry-basics/735049: The angles of elevation of the top of a tower from the points P and Q at distances of 'a' and 'b' respectively from the base of the tower and in the same straight line with it are complementary.What is the height of the tower?1 solutions Answer 449263 by mananth(12270)   on 2013-04-08 01:29:15 (Show Source): You can put this solution on YOUR website!And let Q be at a distance of b from the tower making an angle Let P be at distance a from the tower. This will make an angle Let height of tower be h Tan=h/a Cot =h/a Tan=a/h Tan = h/b
 Polynomials-and-rational-expressions/734990: Multiply and, if possible, simplify. x1 solutions Answer 449261 by mananth(12270)   on 2013-04-08 00:55:44 (Show Source): You can put this solution on YOUR website!x x x x
 Quadratic_Equations/734664: Use the quadratic formula to solve the equation. Give exact answers x^2+2x=81 solutions Answer 449087 by mananth(12270)   on 2013-04-07 08:55:50 (Show Source): You can put this solution on YOUR website!x^2+4x-8 compare with ax^2+bx+c a=1, b=2,c=-8 b^2-4ac= 4 + 32 b^2-4ac= 36 x1=( -2 + 6 )/ 2 x1= 2.00 x2=( -2 -6 ) / 2 x2= -4.00
 Money_Word_Problems/734653: A water trough is constructed by bending a 4- by 6-foot rectangular sheet of metal down the middle and attaching triangular ends (see the figure). If the volume of the trough is 9 cubic feet, find the width correct to two decimal places.1 solutions Answer 449076 by mananth(12270)   on 2013-04-07 01:48:34 (Show Source): You can put this solution on YOUR website!bend in the middle . you will get an isosceles triangle . The height of triangle will be = Area of triangle = (1/2) * base * height A= sqrt(3)/2 * base Length = 6 feet Volume of tank = 6*sqrt(3)/2 * base 9=sqrt(3)/2 * base Base = width of tank = 6sqrt(3)
 Linear-equations/734630: I am really having trouble understanding how to solve a system of equations by graphing. Can someone please help me understand how to set up and work through these types of problems? I do not want to give the problems I have because they are homework, and I want to work my own homework. If you could just provide examples of how to set up, work, and graph I would be grateful. 1 solutions Answer 449073 by mananth(12270)   on 2013-04-07 01:38:23 (Show Source): You can put this solution on YOUR website!-3x-2y = 6 .............1 3x-y = 12 .............2 equation1 when x= 0, y=-3(0,-3) when y = 0,x=-2(-2,0) Equation 2 when x= 0,y=-12 (0,-12) when y =0,x=4(4,0) Plot these points and draw the lines. The point of intersection is the solution of the system. The solution is (2,-6) m.ananth@hotmail.ca
 Volume/734646: a cylindrical water tank near the town library is 15 meters high and has a circumference of 85 meters. what is the APPROXIMATE volume, to the nearest whole number, of the water tank? A) 1,275m^3 B) 2,875m^3 C) 8,627m^3 D) 34,497m^31 solutions Answer 449070 by mananth(12270)   on 2013-04-07 01:23:22 (Show Source): You can put this solution on YOUR website!(85/(2*pi))^2*pi*15= 8624 cm^3 C is the option
 Equations/734608: I have a algebra question that I have been struggling to figure out for a while. It is simplifing a radical equation. The equation is(2√5+3√7)^2. If you can please add the steps so I can work through it to understand it better. Thank you very much!1 solutions Answer 449053 by mananth(12270)   on 2013-04-06 21:58:36 (Show Source):
 Linear-systems/734261: Use a specified method to solve the system of equations. (graphically) y=4-x y=x-6 I understand that the first ordered pair is (-1,4) and the second is (1,-6) I just have no clue how to graph these.1 solutions Answer 448871 by mananth(12270)   on 2013-04-05 22:23:28 (Show Source): You can put this solution on YOUR website!y=4-x y=x-6 Find the x & y intercept points by plugging x=0,y=0 in both equations. equation I when x= 0 y= 4 ,( 0 , 4 ) when y = 0 x= 4 ,( 4 , 0 ) Equation 2 when x= 0 y= -6 ( 0 , -6 ) when y = 0 x= 6 ( 6 , 0 ) You will observe that the point of intersection is 5,-1) The solution of the system is ( 5 ,-1 )
 Volume/734275: suppose that a box has a base with a width of x, a length of x+1, and a height of 2. It is cut from a rectangular sheet of material with an area of 182 inches2 find the dimension of the box. 1 solutions Answer 448868 by mananth(12270)   on 2013-04-05 22:11:36 (Show Source): You can put this solution on YOUR website!width of x, a length of x+1, and a height of 2. It is cut from a rectangular sheet of material with an area of 182 since the height is 2 , 2 inches is cut off from the sides width will be x+4 length will e x+5 area of sheet = (x+4)(x+5)= 182 x^2+9x+20=182 x^2+9x-162=0 x^2+18x-9x-162=0 x(x+18)-9(x+18)=0 (x+18)(x-9)=0 x= 9 which is positive Box dimension 9 by 10 by 2" width = 13" and length = 14" of sheet
 Travel_Word_Problems/734319: a motorist travels 60 miles while driving in the rain. In sunny weather, he drives 21 mph faster and covers 96 miles in the same amount of time. what is the speed in the rain and the speed in sunny weather? Please help me solve this problem.1 solutions Answer 448864 by mananth(12270)   on 2013-04-05 22:00:53 (Show Source): You can put this solution on YOUR website!Time 60 miles in rain = time 96 miles in sun speed in rain = x speed in sun = x+21 time in rain = 60/x Time in sun = 96/(x+21) 60/x= 96/(x+21) 60(x+21) = 96x 60x+1260=96x 36x=1260 x=35 km/h you can continue
 Numbers_Word_Problems/734133: Please walk me through the steps to the following problem. I don't even know how to start. A Ford Explorer can hold a driver and four passengers. A Dodge Caravan can hold a driver and six passengers. The biology dept. is taking a total of 95 people on a field trip. The college wants to use twice as many Caravans as Explorers because the Caravans get better gas mileage. How many of each kind of vehicle are needed?1 solutions Answer 448740 by mananth(12270)   on 2013-04-05 10:13:26 (Show Source): You can put this solution on YOUR website!A Ford Explorer can hold a driver and four passengers. A Dodge Caravan can hold a driver and six passengers. The biology dept. is taking a total of 95 people on a field trip. The college wants to use twice as many Caravans as Explorers because the Caravans get better gas mileage. How many of each kind of vehicle are needed? let explorers be x and caravans be y y=2x 5x+7y=95 substitute y=2x 5x+7(2x)=95 5x+14x=95 19x=95 x=5 5 explorers and 10 caravans
 Travel_Word_Problems/734136: A boat travels 7 km upstream and 7km back. The time for the round trip is 10 hours. The speed of the stream is 4 km/hr. What is the speed of the boat in still water?1 solutions Answer 448739 by mananth(12270)   on 2013-04-05 10:09:15 (Show Source): You can put this solution on YOUR website!A boat travels 7 km upstream and 7km back. The time for the round trip is 10 hours. The speed of the stream is 4 km/hr. What is the speed of the boat in still water? let speed in still water = x km/h with current x+4 km/h against current speed = x-4 km/h time up + time return = 10 7/(x+4)+7/(x-4)=10 multiply equation by (x+4)(x-4) 7(x-4)+7(x+4)= 10(x^2-16) 7x-28+7x+28 = 10x^2-160 14x=10x^2-160 10x^2-14x-160=0 /2 5x^2-7x-80=0 Find the roots of the equation by quadratic formula a= 5 , b= -7 , c= -80 b^2-4ac= 49 + 1600 b^2-4ac= 1649 x1=( 7 + 40.61 )/ 10 x1= 4.76 x2=( 7 -40.61 ) / 10 x2= -3.36 Ignore negative value boat speed 4.761 kph
 Linear-equations/734111: Line through (8,-5) with m=61 solutions Answer 448738 by mananth(12270)   on 2013-04-05 09:59:36 (Show Source): You can put this solution on YOUR website!Equation of line in slope point form y-y1=m(x-x1) (x1,y1)(8,-5) m=6 (y+5)=6(x-8) y+5=6x-48 y=6x-53
 test/734125: A new still unsharpened, wooden pencil has the form of a cylinder. It is 19cm long and has a diameter of 0.8cm.The cylinder hole for the core in the middle of the pencil has a diameter of 0.2cm 1) calculate how much cm3 of wood is needed for such a pencil 1 solutions Answer 448729 by mananth(12270)   on 2013-04-05 09:14:46 (Show Source): You can put this solution on YOUR website!outer volume - inner volume = volume of wood for the pencil outer radius =0.8/2 = 0.4 cm length = 9 cm inner radius = 0.2/2 = 0.1 length = 9 cmm V cylinder = pir^2h V(whole pencil) = pi * 0.4^2*9 V (core) = pi*0.1^2*9 V ( wood) = pi*0.4*2*9-pi*0.1^2*9 = 9*pi(0.4^2-0.1^2) =4.24 cm^3
 Travel_Word_Problems/734095: A train leaves a station at 9.00 p.m. Another train leaves same station at 10.00 p.m. first train runs at 35 kmph and Second train crosses the first train in 4 hours. Find the speed of the second train. 1 solutions Answer 448728 by mananth(12270)   on 2013-04-05 09:05:20 (Show Source): You can put this solution on YOUR website!difference of time between Train I and Train II 1 hour speed of train I = 35 kmph speed of train II = x kmph catch up speed = x-35 catch up time = 4 hours catch up distance = 35 km ( distance traveled in 1 hour by Train I) t= d/s 4 = 35/(x-35) 4(x-35)=35 4x-140=35 4x=140+35 4x=175 x= 43.75 kmph