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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

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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

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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..

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(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

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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

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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)

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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)

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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

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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

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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

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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

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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

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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

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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)

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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)

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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

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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)|

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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

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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}

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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
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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

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.

You can put this solution on YOUR website!
SEE THE FOLLOWING EXAMPLES AND TRY .IF STILL IN DIFFICULTY PLEASE COME BACK
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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)