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# Recent problems solved by 'venugopalramana'

venugopalramana answered: 3288 problems
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 Travel_Word_Problems/39830: After Sam and Pat bike in opposite directions for 2&1/2 hours, they are 80km apart. If Pat's rate is 2 km/h more than Sam's rate, find how far each travels.1 solutions Answer 25263 by venugopalramana(3286)   on 2006-05-30 02:42:45 (Show Source): You can put this solution on YOUR website!After Sam and Pat bike in opposite directions for 2&1/2 hours, they are 80km apart. If Pat's rate is 2 km/h more than Sam's rate, find how far each travels. RELATIVE SPEED WHILE GOING IN OPPOSITE DIRECTIONS =SUM OF SPEEDS= LET SAMS SPEED = S KMPH....PAT'S SPEED IS S+2 KMPH REL.SPEED =S+S+2=2S+2=2(S+1) DISTANCE OF SEPERATION =80 KM TIME =2.5 HRS SO 2.5=80/{2(S+1)}=40/(S+1) S+1=40/2.5=16 S=16-1=15 KMPH....IN 2.5 HRS SAM GOES =15*2.5=37.5 KM P=S+2=17 KMPH.....IN 2.5 HRS PAT GOES =17*2.5=42.5 KM
 Numbers_Word_Problems/37092: Find the value of 'a' such that the graphs of 4y=ax+2 and 1/5y = 1/4x + 7 are parallel. i'm trying to figure out this question cuz my little bro is having problems with it, and i cant get it. thanks1 solutions Answer 25153 by venugopalramana(3286)   on 2006-05-29 07:06:12 (Show Source): You can put this solution on YOUR website!Find the value of 'a' such that the graphs of 4y=ax+2 and 1/5y = 1/4x + 7 are parallel. i'm trying to figure out this question cuz my little bro is having problems with it, and i cant get it. thanks FOR 2 LINES TO BE PARALEL THEIR SLOPES SHOULD BE EQUAL 4Y=AX+2.....OR...Y=AX/4 + 2/4..................I SLOPE=A/4 Y/5=X/4 +7..OR....Y=5X/4+7/4.....................II SLOPE =5/4 A/4 = 5/4 A=4*5/4=5 A=5
 Radicals/37099: This question is from textbook College Algebra A. Is (square root x)^2=x an identity (true for all nonnegitive values of x)? please explain answer B. For the equation x-2sqrtx=0, perform the following: solve for all values of x that satisfies the equation answer & show work C. graph the functions y=x and y=2 sqrt x on the same graph as B.(by plotting points if necessary) Show the points of intersection of these two graphs. Also could you look at this? I think I have it right but I wanna double check A right triangle is a triangle with one angle measuring 90 degrees. In a right triangle, the sides are related by Pythagorean Theorem, c^2=a^2+b^2, where c is the hypotenuse (the side opposite the 90 degree angle). Find the hypotenuse when the other two sides' measurements are 6 feet and 8 feet. so heres what I have: c^2=6^+8^2 c^2=36+64 sqrt c^2= sqrt 100 c= 5 sqrt 41 solutions Answer 25151 by venugopalramana(3286)   on 2006-05-29 06:26:44 (Show Source): You can put this solution on YOUR website!A right triangle is a triangle with one angle measuring 90 degrees. In a right triangle, the sides are related by Pythagorean Theorem, c^2=a^2+b^2, where c is the hypotenuse (the side opposite the 90 degree angle). Find the hypotenuse when the other two sides' measurements are 6 feet and 8 feet. so heres what I have: c^2=6^+8^2 c^2=36+64 sqrt c^2= sqrt 100 c= 5 sqrt 4 CORRECT CONTINUE =5*2=10 IS THE ANSWER..OK..
 Trigonometry-basics/37066: (1) find the exact value of sec(225degrees) (2) In the triangle ABC, find A, if B= 100 degrees, and a=20. (3) In the triangle ABC, find b, iF A=15, C=12 and angle b=150 degrees.1 solutions Answer 25150 by venugopalramana(3286)   on 2006-05-29 06:00:12 (Show Source): You can put this solution on YOUR website! (1) find the exact value of sec(225degrees)=1/COS(225)=1/COS(180+45)=-1/COS(45) =-SQRT2=-1.414 (2) In the triangle ABC, find A, if B= 100 degrees, and a=20. DATA IS INSUFFICIENT.CHECK PROBLEM (3) In the triangle ABC, find b, iF A=15, C=12 and angle b=150 degrees. b^2=a^+c^2-2acCOS(B)=15^2+12^2-2*15*12COS(150)=225+144-30*COS(180-30) =369+30COS(30)=369+30*0.866=369+25.98=394.98 b=SQRT(394.98)=19.87
 Rational-functions/37067: f(x)=3x+2 g(x)=(x-2)/3 1) find [f(g(x))]. show steps. 2) find [g(f(x))]. show steps.1 solutions Answer 25148 by venugopalramana(3286)   on 2006-05-29 05:21:18 (Show Source): You can put this solution on YOUR website!f(x)=3x+2=Y SAY g(x)=(x-2)/3=Z SAY 1) find [f(g(x))]. show steps. WE HAVE TO FIND F(Z)=3Z+2=3{(X-2)}/3 + 2=X-2+2=X 2) find [g(f(x))]. show steps. WE HAVE TO FIND G(Y)=(Y-2)/3={3X+2-2}/3=3X/3=X
 Linear-systems/39698: This question is from textbook College Algebra I can't figure out how to solve this equation 6x-3y=1 -12x+6y=-2 I need to solve using a matrix but I can't figure out how to get the answer in the back of the book. It says the answer is (3y+1/6,y)1 solutions Answer 25142 by venugopalramana(3286)   on 2006-05-29 00:48:03 (Show Source): You can put this solution on YOUR website!I can't figure out how to solve this equation 6x-3y=1 -12x+6y=-2 I need to solve using a matrix but I can't figure out how to get the answer in the back of the book. It says the answer is (3y+1/6,y) THESE ARE DEPENDENT EQNS..HENCE YOU MIGHT BE FACING THE DIFFICULTY HOPE YOU KNOW THE METHOD AUGMENTED MATRIX IS ....COEF.OF X.....COEF.OF Y.....CONSTANT......UNIT MATRIX ............6.......-3..............1...........1..............0 ...........-12..... .6.............-2...........0..............1 WE HAVE TO CONVERT IT NOW.. 1.NEW R1=OLD R1/6 ............1........-3/6...........1/6.........1/6.............0 ............-12........6.............-2..........0..............1 2.NEW R2 =OLD R2+12*R1 ............1.........-1/2...........1/6.........1/6.............0 ............0..........0.............0............2..............1 NOW WE CANNOPT MAKE SECOND ROW ,SECOND COLUMN ELEMENT AS 1 ANY MORE AS IS NORMALLY DONE IN CONSISTENT AND INDEPENDENT EQN. THE PRESENCE OF ZEROS IN ALL 3 COLUMNS IN SECOND ROW INDICATES THAT THE EQN.GIVEN ARE DEPENDENT .SO WE HAVE ONLY ONE INDEPENDENT EQN.IN 2 UNKNOWNS .HENCE WE SHALL NOT HAVE A UNIQUE SOLUTION.WE GET AN INFINITE SET OF ANSWERS .THEY ARE GIVEN BY FIRST ROW WHICH TELLS US THAT 1X-Y/2=1/6 SO TO GET ANY SET OF SOLUTION PUT X ANY VALUE AND FIND Y OR VICEVERSA. SUPPOSE WE PUT X=0...THEN -Y/2=1/6...OR....Y=-2/6=-1/3... SO SOLUTION SET IS (0,-1/3) OR PUT X=1/6....THEN Y=0.. SO SOLUTION SET IS (*1/6,0)....ETC.... ..SAME WAY WITH Y PUT Y=0...THEN X=1/6.... SO SOLUTION SET IS (1/6,0) IN GENERAL IF WE TAKE Y=Y..THEN 1X=Y/2+1/6 = (3Y+1)/6 SO THE SOLUTION SET IS {(3Y+1)/6,Y}..THIS IS WHAT IS GIVEN IN YOUR BOOK. IT MEANS PUT ANY VALUE FOR Y IN THIS SET AND THE RESULTANT NUMBERS WILL GIVE YOU A SOLUTION FOR THE GIVEN EQNS. BUT YOU NEED NOT GIVE THIS ONLY AS AN ANSWER.FOR EX. YOU CAN PUT X=X AND THEN X-Y/2=1/6...SO....Y/2=X-1/6....OR....Y=2X-1/3=(6X-1)/3...HENCE THE ANSWER COULD ALSO BE GIVEN AS {X,(6X-1)/3}...GOT IT?BUT YOU HAVE TO GIVE A GENERAL SOLUTION LIKE THIS AND NOT JUST 1 SET OF VALUES LIKE (0,-1/3)
 Linear_Algebra/39572: Proof of a singular matrix: I need to show (in general) that either matrix A is "singular" or "A^2 = A^(-1)". A is an n*n square matrix such that A^4 = A. I want to know how to do this problem and what happens if the exponent values are different.(for example, odd number exponents?)1 solutions Answer 25103 by venugopalramana(3286)   on 2006-05-28 12:16:02 (Show Source): You can put this solution on YOUR website! I need to show (in general) that either matrix A is "singular" or "A^2 = A^(-1)". A is an n*n square matrix such that A^4 = A. I want to know how to do this problem and what happens if the exponent values are different.(for example, odd number exponents?) PROCEDURE IS TO,TRANSPOSITION TO LEFT OR RIGHT SIDE OF THE EQUATION, RIGHT OR LEFT MULTIPLY BOTH SIDES OF THE EQUATION WITH A OR A^(-1),TAKING COMMON FACTORS ACCORDINGLY....NOTING THAT A^N=A*A*A*...N TIMES AND A*A^(-1)=A^(-1)*A=I...AND A*B=0 IMPLIES A=0 OR B=0..... IN THIS CASE WE HAVE A^4=A (A^4)-A=0 A(A^3-I)=0...HENCE A=0...OR.....A^3-I=0 A^3=I...MULTIPLY WITH A^-1 BOTH SIDES A^2*A*(A^-1)=I*(A^-1)=A^-1 A^2=A^(-1)
 Circles/39617: The ratio of the volume of two spheres is 216:125. What is the ratio of their diameters?1 solutions Answer 25102 by venugopalramana(3286)   on 2006-05-28 12:04:09 (Show Source): You can put this solution on YOUR website!The ratio of the volume of two spheres is 216:125. What is the ratio of their diameters? VOLUME =(4/3)PI*R^3...HENCE VOLUME IS PROPORTIONAL TO CUBE OF RADIUS AND HENCE DIAMETER. SO IF V1/V2=216/125...THEN D1/D2=CUBE ROOT OF (216/125)=6/5
 Circles/39618: The ratio of the volume of two spheres is 216:125. What is the ratio of their diameters?1 solutions Answer 25101 by venugopalramana(3286)   on 2006-05-28 12:03:05 (Show Source): You can put this solution on YOUR website!The ratio of the volume of two spheres is 216:125. What is the ratio of their diameters? VOLUME =(4/3)PI*R^3...HENCE VOLUME IS PROPORTIONAL TO CUBE OF RADIUS AND HENCE DIAMETER. SO IF V1/V2=216/125...THEN D1/D2=CUBE ROOT OF (216/125)=6/5
 Probability-and-statistics/39636: A university football stadium has 1/6 of its students' seats in the end zones. If tickets are randomly selected and mailed to students, what is the probability that a certain student would get end zone seats at all 5 home games?1 solutions Answer 25100 by venugopalramana(3286)   on 2006-05-28 11:59:06 (Show Source): You can put this solution on YOUR website!A university football stadium has 1/6 of its students' seats in the end zones. If tickets are randomly selected and mailed to students, what is the probability that a certain student would get end zone seats at all 5 home games? PROBABILITY=(1/6)^5
 Probability-and-statistics/39637: Suppose that an experiment is repeated four times. A certain event has probability 1/10 in a single repetition of the experiment. What is the probability that it never occurs?1 solutions Answer 25099 by venugopalramana(3286)   on 2006-05-28 11:57:36 (Show Source): You can put this solution on YOUR website!Suppose that an experiment is repeated four times. A certain event has probability OF WHAT ?OCCURRENCE OR NON OCCURRENCE?ASSUMING IT IS FOR OCCURRENCE 1/10 in a single repetition of the experiment. What is the probability that it never occurs? P(O)=1/10=0.1....P(NO)=9/10=0.9 LET US FIND PROBABILITY OF NO OCCURRENCEIN 4 TRIALS =(0.9)^4
 Probability-and-statistics/39638: Suppose that an experiment is repeated four times. A certain event has probability 1/10 in a single repetition of the experiment. What is the probability that it occurs at least once?1 solutions Answer 25098 by venugopalramana(3286)   on 2006-05-28 11:55:30 (Show Source): You can put this solution on YOUR website!Suppose that an experiment is repeated four times. A certain event has probability OF WHAT ?OCCURRENCE OR NON OCCURRENCE?ASSUMING IT IS FOR OCCURRENCE 1/10 in a single repetition of the experiment. What is the probability that it occurs at least once? P(O)=1/10=0.1....P(NO)=9/10=0.9 LET US FIND PROBABILITY OF NO OCCURRENCEIN 4 TRIALS =(0.9)^4 HENCE PROBABILITY OF ATLEAST 1 OCCURRENCE =1-(0.9)^4
 Trigonometry-basics/39644: what is the exact value of 6cos(-61pie/6) - 2sin(13pie/4)1 solutions Answer 25097 by venugopalramana(3286)   on 2006-05-28 11:51:12 (Show Source): You can put this solution on YOUR website!what is the exact value of 6cos(-61pie/6) - 2sin(13pie/4) =6COS(61PI/6)-2SIN{2PI+PI+(PI/4)} =6COS{2*5PI+(PI/6)}-2SIN{PI+(PI/4)} =6COS(PI/6)+2SIN(PI/4) =6*SQRT(3)/2+2*1/SQRT(2) =3*1.732+SQRT(2) =5.196+1.414=6.61
 Trigonometry-basics/39645: what is the general solution to sin(x/2)=(root3/2)1 solutions Answer 25096 by venugopalramana(3286)   on 2006-05-28 11:45:42 (Show Source): You can put this solution on YOUR website!what is the general solution to sin(x/2)=(root3/2) X/2=N*PI+(-1)^N*(PI/3) X=2N*PI+(-1)^N*(2PI/3)
 Linear-equations/39622: This question is from textbook INTERMEDIATE ALGEBRA Find the midpoint of line segment PQ. Q(-3,5) P(-5,-5)1 solutions Answer 25060 by venugopalramana(3286)   on 2006-05-27 12:40:04 (Show Source): You can put this solution on YOUR website!Find the midpoint of line segment PQ. Q(-3,5) P(-5,-5) MID POINT IS AVERAGE OF X COORDINATES AND AVERAGE OF Y COORDINATES (-3-5)/2=-4 AND (5-5)/2=0 (-4,0)
 Sequences-and-series/39635: 4) CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Crane came out and gratefully thanked the traveling salesman for saving his daughter’s life. Mr. Crane insisted on giving the man an award for his heroism. So, the salesman said, “If you insist, I do not want much. Get your checkerboard and place one grain of wheat on the first square. Then place two grains of wheat on the next square. Then place four grains on the third square. Continue this until all 64 squares are covered with grains of wheat.” As he had just harvested his wheat, Mr. Crane did not consider this much of an award, but he soon realized he made a miscalculation on the amount of wheat involved. a)How much wheat would Mr. Brown have to put on the 24nd square? Show work in this space. b)How much total grain would the traveling salesman receive if the checkerboard only had 24 squares? Show work in this space c)Calculate the amount of wheat necessary to fill the whole checkerboard (64 squares). How much wheat would the farmer need to give the salesman? Please provide the answer in either scientific notation, or calculate and show all 20 digits. Answer: 1 solutions Answer 25059 by venugopalramana(3286)   on 2006-05-27 12:37:04 (Show Source): You can put this solution on YOUR website!GOOD PROBLEM .YOU CAN USE IT TO GOOD EFFECT TO PUT ACROSS A PROPOSAL TO YOUR FRIEND ... YOU PROMISS HIM A 100,000..POUNDS EVERY DAY FOR 30 DAYS.LET HIM GIVE YOU IN RETURN JUST A PENNY ON THE FIRST DAY,2 PENNIES ON THE SECOND DAY ,4 PENNIES ON THE THIRD DAY,......ETC FOR 30 DAYS ..AS IN YOUR PROBLEM... I PRESUME HE WILL LAP ON THE PROPOSAL AND I CAN ASSURE YOU THAT YOU WONT REGRET THE PROPOSAL!!!! OK NOW LET US GET BACK TO YOUR PROBLEM AS WELL AS MY SUGGESTION ,WHICH IS JUST A PART OF IT .I AM GIVING BELOW THE MONEY TO BE PAID ON THIS BASIS FOR 30 AND 64 DAYS ,ROUNDED OFF TO POUNDS OR MILLION POUNDS AT THE LATER DAYS. DAY.....MONEY TO BE PAID........MONEY TO BE PAID BY YOUR FRIEND TO YOU IN ............TO YOUR FRIEND..... CENTS...POUNDS...MILLION POUNDS ................POUNDS 1 100000 1 0 0 2 100000 2 0 0 3 100000 4 0 0 4 100000 8 0 0 5 100000 16 0 0 6 100000 32 0 0 7 100000 64 1 0 8 100000 128 1 0 9 100000 256 3 0 10 100000 512 5 0 11 100000 1024 10 0 12 100000 2048 20 0 13 100000 4096 41 0 14 100000 8192 82 0 15 100000 16384 164 0 16 100000 32768 328 0 17 100000 65536 655 0 18 100000 131072 1311 0 19 100000 262144 2621 0 20 100000 524288 5243 0 21 100000 1048576 10486 0 22 100000 2097152 20972 0 23 100000 4194304 41943 0 24 100000 8388608 83886 0 25 100000 16777216 167772 0 26 100000 33554432 335544 0 27 100000 67108864 671089 1 28 100000 134217728 1342177 1 29 100000 268435456 2684355 3 30 100000 536870912 5368709 5 31 100000 1073741824 10737418 11 32 100000 2147483648 21474836 21 33 100000 4294967296 42949673 43 34 100000 8589934592 85899346 86 35 100000 17179869184 171798692 172 36 100000 34359738368 343597384 344 37 100000 68719476736 687194767 687 38 100000 1.37439E+11 1374389535 1374 39 100000 2.74878E+11 2748779069 2749 40 100000 5.49756E+11 5497558139 5498 41 100000 1.09951E+12 10995116278 10995 42 100000 2.19902E+12 21990232556 21990 43 100000 4.39805E+12 43980465111 43980 44 100000 8.79609E+12 87960930222 87961 45 100000 1.75922E+13 175921860444 175922 46 100000 3.51844E+13 351843720888 351844 47 100000 7.03687E+13 703687441777 703687 48 100000 1.40737E+14 1407374883553 1407375 49 100000 2.81475E+14 2814749767107 2814750 50 100000 5.6295E+14 5629499534213 5629500 51 100000 1.1259E+15 11258999068426 11258999 52 100000 2.2518E+15 22517998136853 22517998 53 100000 4.5036E+15 45035996273705 45035996 54 100000 9.0072E+15 90071992547410 90071993 55 100000 1.80144E+16 180143985094820 180143985 56 100000 3.60288E+16 360287970189640 360287970 57 100000 7.20576E+16 720575940379279 720575940 58 100000 1.44115E+17 1441151880758560 1441151881 59 100000 2.8823E+17 2882303761517120 2882303762 60 100000 5.76461E+17 5764607523034230 5764607523 61 100000 1.15292E+18 11529215046068500 11529215046 62 100000 2.30584E+18 23058430092136900 23058430092 63 100000 4.61169E+18 46116860184273900 46116860184 64 100000 9.22337E+18 92233720368547800 92233720369 30 DAY..3000000.........1073741823......10737418.....................11 TOTAL 64 DAY..6400000.........1.84467E+19.....184467440737095000......184467440737 TOTAL YOUR GAIN IN 30 DAYS = 8 MILLION POUNDS..GOT IT BUT ONLY BE CAREFULL THAT YOUR FRIEND WONT RUN OUT OF YOU ON THE 25 TH. DAY.!!! YOUR GAIN IN 30 DAYS = 184467440731 MILLION POUNDS NOW COMING TO THE MATHS PART OF THIS ,THIS SEQUENCE WHERE EACH NUMBER BEARS A CONSTANT RATIO TO ITS PREDECESSOR IS CALLED GEOMETRIC PROGRESSION..HERE YOU FIND EACH NUMBER IS OBTAINED FROM THE PREVIOUS ONE BY MULTIPLYING WITH 2 , CALLED COMMON RATIO. THE LAST NUMBER ON NTH. DAY AND SUM OF SUCH SERIES OF NUMBERS UPTO THE N TH.DAY IS GIVEN BY THE FOLLOWING FORMULAE... N TH. NUMBER = FIRST NUMBER *(C0MM0N RATIO )^(N-1)=A*(R)^(N-1)=(2)^(N-1) IN THIS CASE. SUM UP TO N TH.NUMBER =A*{((R)^N - 1))/(R-1)}={(2)^N-1}IN THIS CASE
 Rational-functions/39601: Please explain to me how this equation can be a function x^2=1+y^21 solutions Answer 25040 by venugopalramana(3286)   on 2006-05-27 06:08:49 (Show Source): You can put this solution on YOUR website!Please explain to me how this equation can be a function x^2=1+y^2 ------------------------------------------------------------------------------- PLEASE FIND BELOW EXPLANATION ON FUNCTION,RELATION AND FUNCTION OF A FUNCTION..FROM GENERAL ENGLISH AND MATHHEMATICALLY SPEAKING. NOW COMING TO F O G IN ENGLISH.LET ME TRY.. FISTLY IT IS READ AS F OF G , WHERE F AND G ARE SYMBOLS FOR DIFFERENT FUNCTIONS. NOW WHAT IS A FUNCTION?IT IS SOME THING LIKE A RELATION-BUT A LITTLE MORE DEMANDING THAN THAT.LET US SEE.. SUPPOSE THE FUNCTION F OF MEANS FATHER OF....SAY JOHN IS FATHER OF EMMA..THEN F OF E MEANING F OF EMMA OR FATHER OF EMMA IN FULL....IS A FUNCTION BECAUSE BESIDES BEING A RELATION IT GIVES US A UNIQUE ANSWER. NAMELY JOHN AND JOHN ONLY ...BUT JOHN'S DAUGHTER IS EMMA IS A RELATION ,AND NOT A FUNCTION BECAUSE THE ANSWER IS NOT UNIQUE...THERE MAY BE ANOTHER TANYA DAUGHTER OF JOHN...SO WE MAY GET MORE THAN ONE ANSWER.THEN IT IS A RELATION BUT NOT FUNCTION.GOT IT.WE GENERALLY USE SYMBOLS,F,G ETC..TO DENOTE FUNCTIONS.IN THE SAME PROBLEM IF WE HAVE TO DEAL WITH MORE THAN 1 FUNCTION AS IS IN THIS PROBLEM,WE USE F & G TO DENOTE 2 DIFFERENT FUNCTIONS.F MAY BE FATHER OF AND G MAY BE MOTHER OF ...ETC... NEXT WHAT IS F OF G ..SAME MEANING ..IT MEANS FATHER OF MOTHER OF SOME BODY SPECIFIED IN THE PROBLEM SAY MARIE...SO IT MEANS FATHER OF MOTHER OF MARIE.THAT IS ALL TO IT. ------------------------------------------------------------------------------ SO IF WE WRITE THE GIVEN EQN.AS Y^2=X^2-1 Y=SQRT(X^2-1)=+ OR - SQRT(X^2-1) THEN WE END UP WITH 2 VALUES OF Y FOR ONE VALUE OF X .SO THE ANSWER IS NOT UNIQUE AND IT WILL NOT BE A FUNCTION. WE CAN MAKE IT A FUNCTION BY DEFINING IT AS Y=|SQRT(X^2-1)|
 Quadratic_Equations/39602: How do you determine the domain and range of the following equation y=|x-2|1 solutions Answer 25037 by venugopalramana(3286)   on 2006-05-27 01:06:51 (Show Source): You can put this solution on YOUR website!How do you determine the domain and range of the following equation y=|x-2| DOMAIN IS THE VALUES X CAN TAKE TO MAKE THE GIVEN EXPRESSION FOR Y,A LEGALLY THAT IS MATHEMATCALLY PERMISSIBLE.CERTAIN OPERATIONS LIKE DIVISION BY ZERO,SQUARE ROOT OF A NEGATIVE NUMBER ETC.. ARE NOT PERMISSIBLE.HERE THERE ARE NO SUCH RESTRICTIONS. HENCE DOMAIN IS ALL REAL NUMBERS . RANGE IS THE CORRESPONDING VALUES Y WILL TAKE.HERE WE FIND THAT Y IS MOD OR ABSOLUTE VALUE OF X-2..HENCE WHETHER X IS 100 OR -100,WE ALWAYS GET Y AS POSITIVE OR ZERO. SO RANGE OF Y IS ALL NON NEGATIVE REAL NUMBERS
 Trigonometry-basics/36996: This question is from textbook College Algebra And Trigonometry Factor the expression sec^2t + 5 sec t + 6 Factor the expression tan^2t - 3 tan t + 101 solutions Answer 24997 by venugopalramana(3286)   on 2006-05-26 12:27:11 (Show Source): You can put this solution on YOUR website!Factor the expression sec^2t + 5 sec t + 6 =SEC^2(T)+3SEC(T)+2SEC(T)+6=SECT{SEC(T)+3}+2{SEC(T)+3}={SEC(T)+3}{SEC(T)+2} Factor the expression tan^2t - 3 tan t + 10... CANNOT BE FACTORISED ..CHECK IF IT IS -10...THEN IT ={TAN(T)-5}{TAN(T)+2}
 Rational-functions/39541: The cost of producing a number of items x is given by C=mx+b, in which b is the fixed cost and m is the variable cost (the cost of producing one more item). (a) if the fixed cost is \$40 and the variable cost is \$10, write the cost equation. (b) Graph the cost equation1 solutions Answer 24992 by venugopalramana(3286)   on 2006-05-26 12:20:46 (Show Source): You can put this solution on YOUR website!The cost of producing a number of items x is given by C=mx+b, in which b is the fixed cost and m is the variable cost (the cost of producing one more item). (a) if the fixed cost is \$40 and the variable cost is \$10, write the cost equation.B=40......M=10 C=10X+40 (b) Graph the cost equation PUT DIFFERENT VALUES FOR X AND FIND C...SAY X=5...C=10*5+40=90...ETC... X....0..........5...........10 C....40.........90..........140
 Rational-functions/39542: Find the slope of the line through the given points. (2,4)and (6,-3)1 solutions Answer 24990 by venugopalramana(3286)   on 2006-05-26 12:16:32 (Show Source): You can put this solution on YOUR website!Find the slope of the line through the given points. (X1,Y1)=(2,4)and (X2,Y2)=(6,-3) SLOPE =(Y2-Y1)/(X2-X1)=(-3-4)/(6-2)=-7/4
 Rational-functions/39544: find the constant of variation k. m varies directly with n; m=144 when n=81 solutions Answer 24988 by venugopalramana(3286)   on 2006-05-26 12:14:29 (Show Source): You can put this solution on YOUR website!find the constant of variation k. m varies directly with n; m=144 when n=8 M=KN...WHERE K IS A CONSTANT... 144=K*8 K=144/8=18
 Quadratic-relations-and-conic-sections/36994: I am trying to come up with an equation for a parabola and I cannot seem to figure it out. I have an algebra 2 project where I have to write a paper on a problem that relates to my life. I want to find out the parabola of a corner kick in soccer. The corner is 25 yards away from the player the ball is being kicked to. The player is 5 foot 6 inches tall. I want the ball to be kicked from the corner to the players head. How could I write an equation for that?1 solutions Answer 24985 by venugopalramana(3286)   on 2006-05-26 12:10:40 (Show Source): You can put this solution on YOUR website!I am trying to come up with an equation for a parabola and I cannot seem to figure it out. I have an algebra 2 project where I have to write a paper on a problem that relates to my life. I want to find out the parabola of a corner kick in soccer. The corner is 25 yards away from the player the ball is being kicked to. The player is 5 foot 6 inches tall. I want the ball to be kicked from the corner to the players head. How could I write an equation for that? STD.EQN.OF PARABOLA TAKING YOUR POSITION AS VERTEX (0,0)AND THE LINE BETWEEN YOU AND THE RECEIVER AS AXIS IS Y^2=4AX THE RECEIVER'S POSITION IS (25*3=75',5.5')IS WHERE THE PARABOLA SHOULD PASS THROUGH...SO 5.5^2=4A*75=300A A=5.5*5.5/300=0.10083333 SO EQN. OF PARABOLA IS Y^2=0.403333X
 Sequences-and-series/39509: Find the sum of the first n terms of the given arithmetic series. a2=-6, a5=-18 (n=8)1 solutions Answer 24983 by venugopalramana(3286)   on 2006-05-26 11:49:49 (Show Source): You can put this solution on YOUR website!Find the sum of the first n terms of the given arithmetic series. a2=-6, a5=-18 (n=8) A2=A+D=-6......................I A5=A+4D=-18.................II 3D=-18+6=-12 D=-4 A=-6+4=-2 SN=(N/2){2*-2+(N-1)(-4)}=(N/2){-4-4N+4}= -2N^2 S8=-2*8^2=-128
 Sequences-and-series/39188: I am not sure if this question is in the proper place, but somebody please help. I do not understand. Using the index of a series as the domain and the value of the series as the range, is a series a function? Include the following in your answer: Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic series? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the geometric series? Give real-life examples of both arithmetic and geometric sequences and series. Explain how these examples might affect you personally.1 solutions Answer 24949 by venugopalramana(3286)   on 2006-05-26 02:17:09 (Show Source): You can put this solution on YOUR website!Using the index of a series as the domain an dthe value of the series as the range, is the series a function? Include in ans: Which one of the basic functions(linear, quadratic, rational, or exponential) is related to the arithmetic series?TN=A+(N-1)*D.............LINEAR EQN. Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the geometric series?TN=A*R^(N-1)............EXPONENTIAL Give real life examples of both arithmetic and geometric sequences and series. Explain how these might affect you personally? REAL LIFE EXAMPLES... ARITHMATIC / PROGRESSION/SERIES....A.P. SUPPOSE YOU ARE SAVING YOUR MONEY IN A BANK BY SYSTEMATIC PLAN OF DEPOSITING 100 \$ A MONTH...THEN STARTING WITH A 100\$ ACCOUNT YOUR MONEY IN CREDIT WITHOUT INTEREST ,WOULD BE AN EXAMPLE OF A.P...IT WILL BE..... TAKING MONTH AS AN INDEX... 100,200,300,400...... GEOMETRIC PROGRESSION/SERIES.....G.P. IN A SIMILAR WAY SUPPOSE YOU DEPOSITED 1000 \$ IN A BANK FOR INTEREST OF 4% PER YEAR,AND THE INTEREST IS CALCULATED AT THE END EVERY YEAR AND ADDED TO THE PRINCIPAL,THEN THE AMOUNT GROWN AT THE END OF YEAR IS AN EXAMPLE OF G.P.TAKING YEAR AS N INDEX....THE AMOUNT AT THE END OF SUCCESSIVE YEARS IS ..... 1000,1000*1.04,1000*1.04^2,1000*1.04^3....ETC... KNOWING THAT A G.P WILL YIELD MORE THAN A.P. ,WE BETTER PUT OUR MONEY IN INTERESTING EARNING SECURITY THAN AT HOME WHERE NO INTEREST ACCRUES.
 Polynomials-and-rational-expressions/39264: Compounded semiannually. P dollars is invested at annual interest rate r for 1 year. If the interest is compunded semiannually, then the polynomial P(1+r/2)^2 represents the value of the investment after 1 year. Rewrite this expression without parentheses. Evaluate the polynomial if P= \$200 and r=10%. I = P(1+r/2)^21 solutions Answer 24948 by venugopalramana(3286)   on 2006-05-26 02:14:23 (Show Source): You can put this solution on YOUR website!Compounded semiannually. P dollars is invested at annual interest rate r for 1 year. If the interest is compunded semiannually, then the polynomial P(1+r/2)^2 represents the value of the investment after 1 year. Rewrite this expression without parentheses. Evaluate the polynomial if P= \$200 and r=10%. IT IS NOT I INTEREST.IT IS AMOUNT INCLUDING PRINCIPAL AND INTEREST= A = P(1+r/2)^2=P[1+R^2/4+2*1*R/2]=P+P*R^2/4+P*R..USING FORMULA (A+B)^2=A^2+B^2+2AB R=10%=0.1 A=200(1+0.1/2)^2=200*1.05^2=220.5
 Square-cubic-other-roots/39405: I'm not sure if I answered this problem right: The square of the first consecutive integer plus the square of the third consecutive integer equals 486 more than the square of the consecutive integer. find the integers. Here's my work: x^2 +x^2 +4X + 4 = 486 + x^2 +2x +1 2x^2 + 4x +4 = 487 + x^2 + 2x Subtract x^2 + 2x + 487 from each side x^2 + 2x - 483 = 0 (x+23) (x-21) x = -23 x=211 solutions Answer 24947 by venugopalramana(3286)   on 2006-05-26 02:08:27 (Show Source): You can put this solution on YOUR website!GOOD TO SEE YOUR ATTEMP.KEEP IT UP!SEE MY COMMENTS BELOW ----------------------------------------------------------------------- I'm not sure if I answered this problem right: The square of the first consecutive integer plus the square of the third consecutive integer equals 486 more than the square of the consecutive integer. find the integers. Here's my work: x^2 + (x+2)^2 = 486 + (x+1)^2 x^2 +x^2 +4X + 4 = 486 + x^2 +2x +1 2x^2 + 4x +4 = 487 + x^2 + 2x Subtract x^2 + 2x + 487 from each side x^2 + 2x - 483 = 0 (x+23) (x-21) x = -23 x=21..EXCELLENT ! NOTHING BETTER COULD BE DONE.KEEP IT UP SO THE INTEGERS ARE 21,22,23 ................OR.......-23,-22,-21
 Expressions-with-variables/39430: In rectangle ABCD, diagonal line BD is one more than twice lined BC. Line CD is 7 more than line BC. find line AB.1 solutions Answer 24946 by venugopalramana(3286)   on 2006-05-26 02:00:45 (Show Source): You can put this solution on YOUR website!In rectangle ABCD, diagonal line BD=D SAY is one more than twice lined BC=W SAY. Line CD=L SAY is 7 more than line BC. find line AB. LET AB=CD=L BC=DA=W HENCE BD^2=D^2=L^2+W^2 WE ARE GIVEN THAT D=2W+1 AND L=W+7...SO (2W+1)^2=(W+7)^2+W^2 4W^2+1+4W=W^2+49+14W+W^2=2W^2+14W+49 2W^2-10W-48=0 W^2-5W-24=0 W^2-8W+3W-24=0 W(W-8)+3(W-8)=0 (W-8)(W+3)=0 W=8=BC AB=L=W+7=15
 Sequences-and-series/39508: I really need some help!! (Fast!!!!) Please.... Using the index of a series as the domain an dthe value of the series as the range, is the series a function? Include in ans: Which one of the basic functions(linear, quadratic, rational, or exponential) is related to the arithmetic series? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the geometric series? Give real life examples of both arithmetic and geometric sequences and series. Explain how these might affect you personally? 2. Use the arithmetic sequence of numbers 2,4,6,8,10...to find the following: a) What is d, the difference between any 2 terms? Ans: Show work below: b) Using the formula for the nth term of an arithmetic sequence, what is 101st term? Ans: c) Using the formula for the sum of an aritmetic series, what is the sum of the first 20 terms? Ans: d) Using the formula for the sum of an aritmetic series, what is the sum of the first 30 terms? Ans: e)What observation can you make about these sums of this series (HINT: It would be beneficial to find a few more sums like the sum of the first 2, then the first 3, etc..)?1 solutions Answer 24945 by venugopalramana(3286)   on 2006-05-26 01:52:13 (Show Source): You can put this solution on YOUR website!Using the index of a series as the domain an dthe value of the series as the range, is the series a function? Include in ans: Which one of the basic functions(linear, quadratic, rational, or exponential) is related to the arithmetic series?TN=A+(N-1)*D.............LINEAR EQN. Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the geometric series?TN=A*R^(N-1)............EXPONENTIAL Give real life examples of both arithmetic and geometric sequences and series. Explain how these might affect you personally? REAL LIFE EXAMPLES... ARITHMATIC / PROGRESSION/SERIES....A.P. SUPPOSE YOU ARE SAVING YOUR MONEY IN A BANK BY SYSTEMATIC PLAN OF DEPOSITING 100 \$ A MONTH...THEN STARTING WITH A 100\$ ACCOUNT YOUR MONEY IN CREDIT WITHOUT INTEREST ,WOULD BE AN EXAMPLE OF A.P...IT WILL BE..... TAKING MONTH AS AN INDEX... 100,200,300,400...... GEOMETRIC PROGRESSION/SERIES.....G.P. IN A SIMILAR WAY SUPPOSE YOU DEPOSITED 1000 \$ IN A BANK FOR INTEREST OF 4% PER YEAR,AND THE INTEREST IS CALCULATED AT THE END EVERY YEAR AND ADDED TO THE PRINCIPAL,THEN THE AMOUNT GROWN AT THE END OF YEAR IS AN EXAMPLE OF G.P.TAKING YEAR AS N INDEX....THE AMOUNT AT THE END OF SUCCESSIVE YEARS IS ..... 1000,1000*1.04,1000*1.04^2,1000*1.04^3....ETC... KNOWING THAT A G.P WILL YIELD MORE THAN A.P. ,WE BETTER PUT OUR MONEY IN INTERESTING EARNING SECURITY THAN AT HOME WHERE NO INTEREST ACCRUES. ------------------------------------------------------------------------------------------ SEE THE FOLLOWING EXAMPLE WHICH IS PRACTICALLY SAME --------------------------------------------------- 1)Use the arithmetic sequence of numbers 2, 4, 6, 8, 10� to find the following: a)What is d, the difference between any 2 terms? Answer: Show work in this space. b)Using the formula for the nth term of an arithmetic sequence, what is 101st term? Answer: Show work in this space. c)Using the formula for the sum of an arithmetic series, what is the sum of the first 20 terms? Answer: Show work in this space. d)Using the formula for the sum of an arithmetic series, what is the sum of the first 30 terms? Answer: Show work in this space. e)What observation can you make about these sums of this series (HINT: It would be beneficial to find a few more sums like the sum of the first 2, then the first 3, etc.)? Answer: 1 solutions -------------------------------------------------------------------------------- Answer 16520 by venugopalramana(1619) on 2006-03-10 07:41:16 (Show Source): 1)Use the arithmetic sequence of numbers 2, 4, 6, 8, 10� to find the following: a)What is d, the difference between any 2 terms? Answer: Show work in this space. D= COMMON DIFFERENCE BETWEEN 2 CONSECUTIVE TERMS 4-2=6-4=8-6=10-8=2....CONSTANT...THIS IS THE PROPERTY OF ARITHMATIC PROGRESSION b)Using the formula for the nth term of an arithmetic sequence, what is 101st term? Answer: Show work in this space. TN=A+(N-1)D=2+(N-1)2=2N T101=2*101=202 c)Using the formula for the sum of an arithmetic series, what is the sum of the first 20 terms? Answer: Show work in this space. SN=(N/2){2A+(N-1)D}=(N/2){2*2+(N-1)2}=(N/2)(2N+2)=N^2+N S20=20^2+20=420 d)Using the formula for the sum of an arithmetic series, what is the sum of the first 30 terms? Answer: Show work in this space. S30=30^2+30=930 e)What observation can you make about these sums of this series (HINT: It would be beneficial to find a few more sums like the sum of the first 2, then the first 3, etc.)? Answer: 1.THE SUM IS A QUDRATATIC IN N. 2.IT IS EQUAL TO THE SUM OF NUMBER OF TERMS AND ITS SQUARE
 Sequences-and-series/39454: Please help. 2) Use the geometric sequence of numbers 1, 3, 9, 27, … to find the following: a) What is r, the ratio between 2 consecutive terms? Show work in this space. b) Using the formula for the nth term of a geometric sequence, what is the 10th term? Answer: the sum of the first 10 terms is: Show work in this space. c) Using the formula for the sum of a geometric series, what is the sum of the first 10 terms? Answer: Show work in this space. 1 solutions Answer 24944 by venugopalramana(3286)   on 2006-05-26 01:44:14 (Show Source): You can put this solution on YOUR website!Please help. 2) Use the geometric sequence of numbers 1, 3, 9, 27, … to find the following: a) What is r, the ratio between 2 consecutive terms? Show work in this space. 3/1=9/3=27/9=3=R=COMMON RATIO b) Using the formula for the nth term of a geometric sequence, what is the 10th term? TN=A*R^(N-1) T10=1*3^(10-1)=3^9 Answer: the sum of the first 10 terms is: Show work in this space. SN=A*{(R^N -1)/(R-1)} S10=1*{(3^10 -1)/(3-1)}=(3^10 -1)/2 c) Using the formula for the sum of a geometric series, what is the sum of the first 10 terms? Answer: Show work in this space.
 Quadratic-relations-and-conic-sections/39221: Identify the conic section represented by the equation by writing the equation in standard form. For a parabola, give the vertex. For a circle, give the center and the radius. For an ellipse or hyperbola, give the center and the foci. Sketch the graph. 16x^2+96x-9y^2+36y=361 solutions Answer 24855 by venugopalramana(3286)   on 2006-05-25 12:33:12 (Show Source): You can put this solution on YOUR website!SEE THE FOLLOWING EXAMPLES AND TRY .IF STILL IN DIFFICULTY PLEASE COME BACK ------------------------------------------------------ I was wondering if anyone could help me determine what type of figure is given, then graph. Show all work and label all key points (asymptotes, foci, vertices, the directrix, the center) where applicable. x^2 – 4x – 8y = 12 1 solutions Answer 22495 by venugopalramana(1898) About Me on 2006-05-07 08:31:50 (Show Source): TIP: 2.IF X^2 OR Y^2 IS ONLY PRESENT ,IT COULD BE A PARABOLA. 2.) x2 – 4x – 8y = 12 COMPLETE SQUARE (X^2-2*2X+2^2)-2^2=12+8Y (X-2)^2=8Y+16=8(Y+2) THIS IS THE EQN.OF A PARABOLA .STD EQN. IS (X-H)^2=4A(Y-K),WHERE (H,K) IS THE VERTEX....(2,-2) HERE. 4A=LATUS RECTUM =8 HERE...A=2 FOCUS IS (H+A,K).....(2+2,-2)=(4,-2)..HERE. DIRECTRIX IS X-H+A=0... X-2+2=0...OR....X=0.. AXIS IS Y-K =0..Y+2=0 graph( 500, 500, -20, 20, -20, 20,(x^2-4*x-12)/8 ) Quadratic-relations-and-conic-sections/36550: I was wondering if anyone could help me determine what type of figure is given, then graph. Show all work and label all key points (asymptotes, foci, vertices, the directrix, the center) where applicable. 9x^2 – 96y = 16y^2 + 18x + 279 1 solutions Answer 22493 by venugopalramana(1898) About Me on 2006-05-07 08:27:57 (Show Source): TIP IF X^2 AND Y^2 HAVE DIFFERENT COEFFICIENTS WITH OPPOSITE SIGNS THEN IT COULD BE HYPERBOLA 3.) 9x2 – 96y = 16y2 + 18x + 279 COMPLETE SQUARE... {(3X)^2-2*3X*3+3^2}-3^2-{(4Y)^2+2*4Y*12+12^2}-12^2=279 (3X+3)^2-(4Y+12)^2=279+9+144=432 9(X+1)^2-16(Y+3)^2=432......NOW DIVIDE THROUGH OUT WITH 432 TO GET 1 ON THE RHS. (X+1)^2/(432/9) -(Y+3)^2/(432/16)=1 (X+1)^2/48 - (Y+3)^2/27 =1 THIS THE EQN. OF A HYPERBOLA.STD.EQN.IS. (X-H)^2/A^2 - (Y-K)^2/B^2=1..WHERE (H,K) IS CENTRE.....(-1,-3) HERE TRANSVERSE AXIS IS Y=K...Y=-3 LENGTH OF TRANSVERSE AXIS=2A.. ...=2SQRT(48) CONJUGATE AXIS IS X=H.......X=-1 LENGTH OF CONJUGATE AXIS = 2B =2SQRT(27) ECCENTRICITY=E=SQRT{(A^2+B^2)/A^2} =SQRT{(48+27)/48}=SQRT(75/48) A*E=SQRT(48)*SQRT(75/48)=SQRT(75) FOCI ARE (H+-AE,K)......(-1+SQRT(75),-3) AND ......(-1-SQRT(75),-3) A/E=SQRT(48)/SQRT(75/48)=48/SQRT(75) DIRECTRIX ARE X=H+-A/E.... X=-1+48/SQRT(75)...AND X=-1-48/SQRT(75) ASYMPTOTES ARE GIVEN BY (X-H)^2/A^2 = (Y-K)^2/B^2 OR (X-H)/A=+(Y-K)/B AND............(X+1)/SQRT(48) =(Y+3)/SQRT(27) (X-H)/A=-(Y-K)/B.............(X+1)/SQRT(48) = -(Y+3)/SQRT(27) GRAPH IS GIVEN BELOW.. graph( 500, 500, -50, 50, -50, 50, -3+27*(((x+1)^2-48)/48)^0.5,-3-27*(((x+1)^2-48)/48)^0.5) -------------------------------------------------------- Quadratic-relations-and-conic-sections/36551: I was wondering if anyone could help me determine what type of figure is given, then graph. Show all work and label all key points (asymptotes, foci, vertices, the directrix, the center) where applicable. 4(x-1)2 = 4-y^2 1 solutions Answer 22486 by venugopalramana(1898) About Me on 2006-05-07 06:33:13 (Show Source): TIP: IF X^2 AND Y^2 HAVE DIFFERENT COEFFICIENTS OF SAME SIGN ,THEN IT COULD BE ELLIPSE. 4.) 4(x-1)2 = 4-y2 I HOPE IT IS 4(X-1)^2+Y^2=4...DIVIDE WITH 4 (X-1)^2/1^2+Y^2/2^2=1 THIS THE EQN.OF AN ELLIPSE.STD.EQN. IS (X-H)^2/A^2 +(Y-K)^2/B^2=1 WHERE (H,K) IS CENTRTE....(1,0).HERE LENGTH OF MAJOR AXIS IS 2B...2*2=4 MINOR AXIS IS ALONG Y=K....Y=0 LENGTH OF MINOR AXIS IS 2A...2*1=2 ECCENTRICITY=E=SQRT{(B^2-A^2)/A^2} =SQRT(4-1)/1=SQRT(3) B*E=2SQRT(3) B/E=2/SQRT(3) FOCI ARE (H,K+BE) AND (H,K-BE) (1,2SQRT(3)) AND (1,-2SQRT(3)) DIRECTRIX ARE.Y=K+B/E AND K-B/E Y=2/SQRT(3)..AND...Y=-2/SQRT(3) GRAPH IS GIVEN BELOW... graph( 500, 500, -3, 3, -3, 3, (4-4*(x-1)^2)^0.5,-(4-4*(x-1)^2)^0.5)