Saturday, August 2, 2014

Find a and b if 25x^2+a+36y^2 and 9x^4/25-b+25x^2/9 are squares.

If the sum 25x^2+a+36y^2 represents a perfect square,
we'll apply the formula:


(u + v)^2 = u^2 + 2uv +
v^2


We notice that the missing term is 2uv =
a.


We'll identify u^2 = 25x^2 => u = sqrt 25x^2
=> u = 5x


v^2 = 36y^2 => v = sqrt 36y^2
=> v = 6y


25x^2 + a +
36y^2


2uv = 2*5x*6y


2uv =
60xy


a = 60xy


The missing term
in the quadratic expression is 60xy and the completed square will
be:


(5x+6y)^2 = 25x^2 + 60xy +
36y^2


4) We notice that the missing term is
b = -2uv from the formula:


(u - v)^2 = u^2 - 2uv +
v^2


We'll identify u^2 = 9x^4/25 => u = sqrt 9x^4/25
=> u = 3x^2/5


v^2 = 25x^2/9 => v = sqrt
25x^2/9 => v = -5x/3


9x^4/25 - b +
25x^2/9


-2uv =
-2*3x^2*5x/5*3


-2uv =
-2x^3


The missing term in the quadratic expression is b =
-2x^3 and the completed square will
be:


(3x^2/5 - 5x/3)^2 = 9x^4/25- 2x^3 +
25x^2/9

(x-9)^1/4-(x+2)^1/3=-1 Find x !

For the beggining, we'll impose constraint of existence of
radical (x-9)^1/4.


x - 9 >=
0


x >= 9


So, the
solutions of the equation have to be located in the range [9 ;
+infinite).


NOw, to solve the equation, we'll note a =
(x-9)^1/4 and b = (x+2)^1/3.


We'll raise a to the 4th
power:


a^4 = x - 9 (1)


We'll
raise b to the 3rd power:


b^3 = x + 2
(2)


We'll subtract (2) from (1) and we'll
get:


a^4 - b^3 = x - 9 - x -
2


a^4 - b^3 = -11 (3)


a - b =
-1 => b = a + 1


We'll substitute b in
(3):


a^4 - (a+1)^3 = -11


a^4 -
a^3 - 3a^2 - 3a - 1 = -11


a^4 - a^3 - 3a^2 - 3a + 10 =
0


We'll factorize and we'll
get:


(a-2)(a^3 +a^2 - a - 5) =
0


a^3 +a^2 - a - 5 is
irreducible


a - 2 = 0


a =
2


(x-9)^1/4 = 2


x - 9 =
16


x = 16+9


x =
2


25 is located in the range [9 ;
infinite).


We'll accept it as
solution of the equation, so x = 25.

Friday, August 1, 2014

Prove the following identity: (csc^2) x = (2 sec 2x) / (sec 2x - 1)

We have to  prove that (csc x)^2 = (2 sec 2x) / (sec 2x -
1)


(2 sec 2x) / (sec 2x -
1)


=> [2*(1/ cos 2x)] / [((1/ cos 2x) -
1)]


=> [2*(1/ cos 2x)] / [((1 - cos 2x)/ cos
2x)]


=> 2/ (1 - cos
2x)


=> 2 / ( 1- (1 - 2 (sin
x)^2))


=> 2 / [ 1 - 1 + 2(sin
x)^2]


=> 2 / 2 (sin
x)^2


=> cosec x


This
proves that the right hand side = left hand
side


Therefore we prove that (csc x)^2 = (2
sec 2x) / (sec 2x - 1)

In the novel of Fahrenheit 451, how do the characters, Clarisse, Faber, the unidentified woman, and Granger influence Montag? Use quotes.I NEED...

Montag is heavily influenced by each one of those
characters.


Clarisse: Just
talking with Clarisse stimulates Montag's thought processes.  She is honest and
straightforward and Montag says he can see himself in her eyes.  "How rarely did other
people's faces take of you and throw back to you your own expression, your own innermost
trembling thought?" (pg 11) She makes him aware that he never stops to think.  He tells
Mildred that Clarisse was "the first person I can remember who looked straight at me as
if I counted."( pg 72) She starts his
transformation.


Faber: Montag
actually met Faber before he met Clarisse. On page 10 Montag thinks that he hasn't had
such a strange experience (Clarisse) since the night he met an old man in the park. 
That old man was Faber. When Montag realizes that he wants to read books, he thinks"But
where do you get help, where do you find a teacher this late?" (pg 74)  Faber is that
teacher.  After he talks with Faber he knew that he was two men: Montag who knew nothing
and Faber who would educate him through his earpiece.  He would be Montag + Faber and he
would become a fine "wine" or a better, educated
person.


Unidentified Woman:
This woman had a tremendous impact on Montag.  She stayed in her burning
house  and burned with her books.  It made Montag realize "There must be something in
books, things we can't imagine to make a woman stay in a burning house, there must be
something there.  You don't stay for nothing." (pg 51).  He also realized that "It took
some man a lifetime maybe to put some of his thoughts down, looking around at the world
and life, and then I come along in two minutes and boom! it's all over." (pg 52) The
night Montag saw this woman die with her books was the last night that Montag burned
books.


Granger: Granger
totally changes Montag's plans for fighting the system.  When Montag and Faber planned
to rebel, they were going to run off some books, plant them in firemen's homes, and then
burn the homes.  They were going to make the firemen look treasonous and try to close
the firestations.  Granger has a different plan.  Each person memorizes a
separate book.  "All we want to do is keep the knowledge we think we will need intact
and safe.  We're not out to incite or anger anyone yet. for if we are destroyed, the
knowledge is dead, perhaps for good." (pg 152) He knows that they cannot save all the
books or even memorize two books.  Each man is responsible for the information in one
book.  He calls the men "nothing but dust jackets for books" (pg
153)


The pages may not be exact depending on the edition of
the book you are reading.  However, the information will be found in the immediate
vicinity.

How was India discovered?

People in India have been living for more than 50,000
years. There was really no need for India to be discovered by the natives of India, at
least not for the last 50.000 years.


Also India has been a
well known country among many other countries for many thousand years. How and when
people in each of these countries discovered India is not known to historians. All that
is known is that India started trading with other countries many thousand years back.
Indications of India's trading links are available in its ancient books. For example
Mahabharata the biggest epic in the world written many thousand of
years back, has mentions of many countries and the goods that India imported from
them.


If you are talking about discovery of sea route from
Europe to India, this was discovered by Vasco de Gama in 1498.

In The Great Gatsby, in what way is Gatsby the personification of the American dream?Also, how does Gatsby reflect the story of America and its...

Gatsby personifies the American Dream in two ways, in its
purity and in its eventual corruption. As a young man, Jimmy Gatz of North Dakota
dreamed of the future; he wanted to free himself from grinding poverty, achieve
important accomplishments, and make a new, better life for himself. As his list of
Resolves and daily schedule indicate, most of his day was spent working; the remainder
of his time was spent in activities to make himself stronger, healthier, more educated,
and successful. He also tried to become a better person. One of the notations in his
journal of that time says, "Be better to parents." Almost none of his time was set aside
for entertainment. He was hard-working, disciplined, focused, and motivated to succeed.
In these respects, he can be seen to represent all those who reached for the American
Dream, believing they could attain it through their own
efforts.


Jimmy Gatz does make his way out of North Dakota,
and as Jay Gatsby he rises out of poverty and builds a spectacular fortune, almost
overnight--the American Dream realized. However, he accomplishes no great deeds, builds
nothing of value, and his success does not develop from hard work or creativity. He
embraces a corrupt lifestyle and becomes corrupt himself--a criminal who works for a
gangster. By the 1920s, the novel suggests, great wealth can be achieved by only a
few--those who inherit it or those who acquire it as famous entertainers, athletes, or
criminals. No matter hard the honest George Wilson works, for example, he will never
escape his poverty. The purity of the American Dream as it once existed is over,
destroyed by social class and social corruption. Even baseball, the great American sport
born in the 19th Century, has become corrupt.


The East/West
motif and symbolism in the novel lie at its heart. Gatsby's early association with Dan
Cody brings in elements of American history, with Cody representing the century gone by.
The days of the 19th Century in which a pioneer like Cody could challenge the frontier
and build a fortune are over, and with the death of the frontier, America changed
forever; the best had been lost. Nick's contemplation of the early Dutch settlers at the
conclusion of the novel emphasizes this theme.


The only
part of America that somehow remains exempt from the corruption of the new century is
Nick Carraway's Midwest. His memories of the Midwest of his youth collide with the
hollow, selfish decadence he observes in the East; his Midwestern values conflict with
the soulless amorality he finds in the Buchanans and their social class. Appalled, he
turns his back on the East and goes home to the
Midwest.


Nick decides that all he has experienced has been
"a story of the West." He, Gatsby, and even Tom and Daisy originated in the West and all
came to the East; Gatsby is destroyed and Nick, because he is a person of integrity,
cannot stay. Only Tom and Daisy belong there because they had been corrupted by their
inherited wealth before they arrived.

How does Johnson defend Shakespeare's mixing of tragic and comic elements?

Samuel Johnson's Preface to
Shakespeare
is a monumental work in the world of literary criticism, and his
explanation for Shakespeare's tendency to include tragedy in his comedies and comedy in
his tragedies demonstrates that Johnson understood and admired the bard's ability to
depict human nature in a realistic fashion.  Johnson wrote that human
nature


readability="8">

“partakes of good and evil, joy and
sorrow, mingled with endless variety of proportion and innumerable modes of
combination.”



Thus,
Shakespeare's inclusion of comedic relief in his tragedies or tragic figures (Shylock in
Merchant of Venice) is simply the playwright's accurate portrayal
of life's true nature--it is a blend of hardship and
happiness, which often exists in one's life at the same time.

Calculate tan(x-y), if sin x=1/2 and sin y=1/3. 0

We'll write the formula of the tangent of difference of 2 angles. tan (x-y) = (tan x - tan y)/(1 + tan x*tan y) ...