If a:b= 5:6 and b:c= 3:7, what is the value of (3c+2a):a? Express your answer as a common fraction.

Given a:b = 5: 6 and b:c =
3:7.

We have to find the value of
(3c+2a):a.

Solution:

We
rewrite fractions as below:

a/b = 5/6. So a =
5b/6

b/c = 3/7. So  b = =
3c/7.

Therefore  a = 5b/6 = (5/6)(3c/7) =
5c/7.

Thus we expressed a = 5c/7 and  b = 3c/7 in terms of
c.

Thererore (3c+2a):a = {3c+2*5c/7}:(5c/7) . We multiply
both terms on the right side by 7.

=> (3c+2a):a =
7(3c+2*5c/7): (5c)

=> (3c+2a):a = (21c+10c):
5c

=> (3c+2a):a = (31c):5c. We divide the terms on
the right by c.

=> (3c+2a):a = 31:5. Or (3x+2a)/a =
31/5 .

So (3c+2a)/a =
31/5.

Why does Athena end the epic with unanswered questions?This is in book 24, The Odyssey by homer, Translated by Robert Fitzgerald.

I have looked at three different versions of Homer's
The Odyssey, Book 24, and have found questions by Athena (Athene)
to her father Zeus, but they are not unanswered, at the end of the book. Fitzgerald's
translation is not available on line that I can find, but the three translations I have
read tell the same tale.

Athena's questions are, based on
www.ellopos.net, are as follows:

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Now Athene spake to Zeus, the son of Cronos,
saying: 'O Father, our father Cronides, throned in the highest, answer and tell me what
is now the hidden counsel of thy heart? Wilt thou yet further rouse up evil war and the
terrible din of battle, or art thou minded to set them at one again in
friendship?'

In essence,
perhaps Athena is looking for her father's guidance, approval, or both. He is, after
all, father to all of the gods, and to mankind itself (according to Greek mythology).
Perhaps she also realizes that at this point of Odysseus' very long journey home from
the Trojan War, that Zeus' help might be necessary to guarantee that Odysseus' trials
finally come to an end.

Zeus responds to his daughter,
reminding her that she has been leading the affairs of Odysseus,
guiding him and interceding for him for all these years. Zeus does
offer advice that the "better" way to end the fighting that is threatened by the family
members of the unwelcome suitors Odysseus killed is try to make get both sides to form a
covenant (promise) to bring harmony between them so that Odysseus may reign in peace for
the remainder of his days.

Athena does
assist Odysseus in reaching this goal, while also minimizing further
bloodshed.

Why do you think Shakespeare gives Caesar so few lines and so little stage time, even during the scene in which he appears in Julius Caesar?

Julius Caesar simply isn't a main character in
Shakespeare's Julus Caesar.  Brutus is the tragic figure of the
play, and the central conflict is between Brutus and Antony.  Caesar is simply the focal
point of the conflict between Brutus and Antony.

Actually,
in fact, Caesar's assassination is the focal point, not even Caesar
himself.

Also, the question of whether or not Caesar would
have become a tyrant if crowned emperor must remain ambiguous, or up in the air. 
Shakespeare was probably limited in how much he could reveal about Caesar.  The
reader/viewer must not have a definite idea about Caesar's future.  If Caesar is
characterized as either definetely a tyrant, or definitely not a tyrant, a clear villain
emerges.  If he is definitely tyrant material, Brutus and the conspirators do the moral
and necessary thing by assassinating him, and Antony is a clear villain.  But if
Caesar is definitely not tyrant material, then Brutus is clearly evil and ignoble
(instead of noble) and clearly a villain. 

In other words,
if Caesar is thoroughly characterized the play is not a tragedy and Brutus is not a
tragic figure.  Caesar has to be a little realized figure for the play to work as it
does.

Find the points on the graph of y= 1/3 x^3 - 5x - 4/x at which the tangent is horizontal.

Given the curve y= (1/3)x^3 - 5x -
4/x

We need to find the horizontal tangent line to the
curve y.

==> The equation of the line
is:

y-y1 = m(x-x1) where m is the
slope.

But we know that the horizontal line has a slope of
zero.

==> y= y1 ( y1 is the point of tendency ( 0,
y1)

Now we will differentiate the curve
y.

==> y' = x^2 - 5 +
4/x^2

Now we will find the point of tendency where the
slope is zero.

==> x^2 - 5 + 4/x^2 =
0

==> We will multiply by
x^2

==> x^4 - 5x^2 + 4 =
0

Now we will
factor.

==> (x^2 -4)(x^2 -1) =
0

==> (x-2)(x+2)(x-1)(x+1) =
0

Then the point where the curve has tangent line
are:

x1= 2 ==> y1= 8/3 - 10 - 2 =
-28/3

x2= -2  ==> y2 = -8/3 +10 +2 =
28/3

x3= 1==> y3= 1/3 -5 -4 =
-26/3

x4= -1 ==> y4 = -1/3 + 5 +4 =
26/3

Then the points where the tangents are
horizontal are:

(2,-28/3) ,
(-2, 28/3), (1, -26/3) , and (-1, 26/3)

Write the fraction (3x-2)/(x-3)(x+1) as partial fractions.

We'll decompose the rational function in partial
quotients:

(3x-2)/(x-3)(x+1) = A/(x-3) +
B/(x+1)

We'll calculate LCD of the 2 ratios from the right
side.

The LCD is the same with the denominator from the
left side.

LCD =
(x-3)(x+1)

We'll multiply both sides by (x-3)(x+1) and the
expression will become:

(3x-2) = A(x+1) +
B(x-3)

We'll remove the
brackets:

3x - 2 = Ax + A + Bx -
3B

We'll combine like terms form the right
side:

3x - 2 = x(A+B) +
(A-3B)

We'll compare and we'll
get:

3 = A+B

-2 = A -
3B

We'll use the symmetric
property:

A+B = 3 (1)

A - 3B =
-2 (2)

We'll multiply (1) by
3:

3A+3B = 9 (3)

We'll add (3)
to (2):

3A+3B+A - 3B =
9-2

We'll eliminate like
terms:

4A = 7

We'll divide by
4:

A = 7/4

We'll substitute A
in (1):

A+B = 3

7/4 + B =
3

We'll subtract 7/4 both
sides:

B = 3 - 7/4 => B = (12-7)/4 => B =
5/4

The fraction (3x-2)/(x-3)(x+1) = 7/4(x-3)
+ 5/4(x+1), where 7/4(x-3) and 5/4(x+1) are partial
fractions.

Write a note on the description of hell in Milton's Paradise Lost.

Milton's Paradise Lost is filled with
very imagery in all of its many books, but Milton's descriptions of hell are especially
vivid, and keep in line with our general understanding of hell as being a place of fire
and punishment.

In Book 1, Milton describes what happened
to the fallen angels who dared to challenge God in Heaven.  He states that the angels
were all

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hurled headlong flaming from the ethereal sky /
With hideous ruin and combustion down / To bottomless perdition, there to dwell / In
adamantine chains and penal fire ... his (Satan's) horrid crew / Lay vanquished, rolling
in the fiery gulf.

This first
description of hell is very clearly a place of terror and torment.  The rebel Angels
were thrown from the beautiful sky of heaven down to an unending hell of damnation. 
There they are suffering in the fires that they cannot escape
from. 

From there, the descriptions goes on to reinforce
the above mentioned description.  Hell is described as
a

dismal
situation waste and wild / A dungeon horrible, on all sides round / As one great furnace
flamed, yet from those flames / No light but rather, darkness
visible. 

This description is
especially interesting in the final image.  Normally we think of fire and picture the
warm lighted glow that is emitted from the flames, but this fire is so intense and
other-worldly no light comes forth.  It is actually darker than seems possible.  It is
in incredibility frightening description.  From there, the description continues with
interesting and powerful word choices and short phrases.  There is a mention of the
"fiery deluge" which suggests a flood of fire -- a flood is usually thought to be
overwhelming and unstoppable.   When Beelzebub tries to rally the angels to be strong in
the midst of this torture he acknowledges the "dreary plain" that is "forlorn and
wild."  He calls it a "seat of desolation" and describes the flames as "livid."  That is
an interesting word choice because the reader might expect "vivid" meaning bright and
lively, but he uses "livid" to draw the connotation of anger and power.  Even though
this hell is an awful and frightening place, Satan wants his followers to "toss off the
fiery waves" and overcome this "dire calamity."  He rallies the other angels to try to
rise from the firey pit they are in and to embrace the idea that even though they are
damned

The
mind is its own place, and in itself / Can make a Heaven of Hell, or a Hell
of Heaven. 

 The angels may
be eternally in this place, but they can still have great influence in the world.  They
can embrace the idea that they are "in charge" of hell and no longer have God in charge
of them.  The rest of Paradise Lost is about how Satan sets about to get his revenge on
God.  As we know from the Bible, he sees his opportunity in the characters of Adam and
Eve in God's Garden of Eden.  As they are brought down by sin, so is all of
humanity.

if you're standing on ice-skates and throw a basketball forward. how your motion after you throw it compares with the motion of the basketball?

To determine your motion when you throw a basketball while
you are standing on ice-skates; we use the principle of conservation of total
momentum.

Let us assume the
following:

your mass: M

the
mass of the ball: Mb

the velocity of the ball when it is
thrown: Vb in the horizontal direction

your velocity:
V

We have assumed that the ball is thrown in a horizontal
direction. The total momentum of the system which includes you and the ball was 0
initially, after you threw the ball it became Vb*Mb -
V*b

So we arrive at Vb*Mb - V*b =
0

=> V = Vb*Mb / M

This
means you are pushed back with a velocity equal to the product of the mass of the ball
and the velocity of the ball divided by your mass.

The ball
is pulled downward by the gravitation force of attraction equal to Mb*g after it is
released. If it was held at a height of d when it was thrown, the time taken by the ball
to strike the ground is sqrt (2*d/g). During this duration the ball has moved in the
horizontal direction by a distance Vb*sqrt (2*d/g).

You
follow a horizontal path after the ball is released, whereas the ball initially follows
a parabolic path downwards. If we take the force of friction to be negligible, you
continue to move in a straight line, and the ball after it strikes the ground starts to
move in a straight horizontal line too.