geno3141

Since there is a vertical asymptote of x = 3, there will be a factor of (x - 3) in the denominator.

This makes c = -3.

Since there is a horizontal asymptote of y = - 4, the ratio ax/x must be -4.

This makes a = -4.

Since there is an x-intercept at (1,0), we can look at  y = (-4x + b) / (x - 3)

and substitute 1 for x and 0 for y:  0  =  (-4·1 + b) / (0 - 3).

This makes b = 4.

f(x)  =  (-4x + 4) / (x - 3)

To get the answer to b, multiply both the numerator and denominator by 2.

The asymptotes will intersect at the center of the hyperbola.

The center of the hyperbola is (1,2).

The slope of the asymptotes will be found by the square roots of the denominators, so the

slopes are  8/6  and  -8/6.

Equations:  y - 2  =  (4/3)(x - 1)     and     y - 2  =  (-4/3)(x - 1)

Let's assume that four of the players are in the left bracket and the other four players are in the

right bracket.

The number one ranked player always makes it to the finals. The other players, depending upon

the draw, that could make it to the finals are players number two, three, four, and five.

Let's assume that player number one is in the left bracket (the analysis is the same if this 

player is in the right bracket) ...

Where is player number two? 

If this player is in the right bracket, this person will make it into the finals; if not, this player will

be defeated by player number one.

After player number one is put into the left bracket, there are seven slots open for player number

two; four of them in the right bracket (and makes it into the finals and three in the left bracket and

loses to player number one).

The probability that this player is in the right bracket is 4/7.

We need to multiply this probability by the rank:  (4/7) x (2)  =  8/7  =  1.143.

If both player number one and player number two are in the left bracket, where is player number three?

There are six slots left and four are in the right bracket, so the probability that this player is in the right

bracket is 4/6.

The probability is (3/7) x (4/6)  =  12/42  =  2/7

[We need to include 3/7 because we have to have player number two in the left bracket.]

Multiplying this probability by the rank:  (2/7) x (3)  = 6/7  =  0.857

If player numbers one, two, and three are all in the left bracket, where is player number four? 

The probability that players one, two, and three are all in the left bracket and player number four

is in the right bracket is (3/7) x (2/6) x (4/5)  =  4/35.

Multiplying this probability by the rank:  (4/35) x 4  =  0.457

If players number one, two, three and four are all in the left bracket, player number five will be

in the right bracket:  (3/7) x (2/6) x (1/5) x (4/4)  =  1/35

Multiplying this probabiliy by the rank:  (1/35) x 5  =  0.143

Adding these answers together:  1.143 + 0.857 + 0.457 + 0.143  =  2.6

Bottom side:  rectangle =  29 x 30

Top side:        rectangle = 12 x 30

Right side:     rectangle = 30 x 26

Left side:       rectangle = 30 x 25

Front side:     trapezoid = ½ x 24 x (29 + 12) 

Back side:     trapezoid = ½ x 24 x (29 + 12) 

Calculate the individual areas and add them together.

HInt:  arc(MN) + arc(JK)  =  angle(MLN) + angle(JLK)

--->                    arc(MN)  =  angle(MLN) + angle(JLK) - arc(JK)

Put in the numbers and finish ...

You can use Pick's formula:

     Area  =  (Number of Boundary Points) / 2 + (Number of Interior Points) - 1

The top figure:  Number of Boundary Points = 10

                         Number of Interior Points = 2

--->   Area  =  (10/2) + 2 - 1  =  5 + 2 -1  =  6

This formula can also be used for the other two figures resulting in the same answers as given by the other person.

First, factor the denominator:  x² + 6x - 7  =  (x + 7)(x - 1).

Now, the function becomes:  (x - 1) / (x² + 6x - 7)   --->   (x - 1) / [ (x + 7)(x - 1) ]

The "x - 1" term in the denominator cancels the "x - 1" term in the numerator; this means that there 

will be a "hole" in the graph at the point where x = 1.

The function has become:  1 / (x + 7)

There will be a vertical asymptote at the point where x = -7.

There is one asymptote.

Work from the inside out:  first find the inverse of 3 and then find the inverse of that answer.

Since  f(4) = 3,  f^-1(3) = 4

Since f(7) = 4,  f^-1(4) = 7.

Andy      =   12 pieces

Bert        =    4 pieces

Charles  =    4 pieces

Nuts         =    4 pieces

Plain        =    8 pieces

Caramel  =    8 pieces

Give Bert the 4 with nuts.

Give Charles 4 with caramel.

Give Andy the 8 plain and 4 with caramel.

Hint:  f( f^-1(a) )  =  a.