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Side   a = sqrt(pi² / 2)  

simplify:     cos( 2 · arcsec(2) )

Since  sec = 1 / cos  and  cos  =  1 / sec

arcsec( 2)   =  arccos( 1/2 )  =  60^o

cos( 2 · arcsec(2) )  =  cos( 2 · 60^o )  =  cos( 120^o )  =  .......

To multiply  r₁·cis( theta₁ )  x  r₂·cis( theta₂)  =  r₁·r₂·cis( theta₁ + theta₂ )

[multiply the r-values and add the angles]

  √3/2cis(2π/9)·√6/4cis(3π/2)  =  ( 3/2 )·( √6/4 ) cis( 2π/9 + 3π/2 )

                                               =  ......

1.In parallelogram ABCD, A = (4, 5), B = (1, 4), and C = (6, 2). Find the coordinates of D.

We  have  three possibilities  for  D

a)  ( 4 + 1  - 6, 5 + 4 - 2)   =   ( -1, 7)

b) ( 4 + 6 - 1, 5  + 2 - 4)  =   ( 9, 3)

c)  ( 1 + 6 - 4, 4 + 2 - 5)  =  ( 3, 1)

We need to add the same amount to 20, 50, and 100 to get a geometric sequence.

Call this amount 'x".

Then our terms will be   t₁  =  20 +x,   t₂  =  50 + x,   and   t₃  =  100 + x

To be a geometric sequence, the terms  t₁,  t₂,  and  t₃  must fit in this proportion:

    t₃ / t₂  =  t₂ / t₁

Substitutng:            (100 + x) / (50 + x)  =  (50 + x) / (20 + x)

Cross-multiplying:  (100 + x) · (20 + x)  =  (50 + x) · (50 + x)

Multiplying out:         2000 + 120x + x²  =  2500 + 100x + x² 

Simplifying:                                    20x  =  500

                                                          x  =  25

So, the numbers are  45,  75,  and  125

The ratio:  125 / 75  =  5/3

and              75 / 45  =  5/3

xy  = 6 - 2x - 3y                     xz  =   30  - 4x   - 3z

xy + 3y  =  6 - 2x                   xz + 3z  =  30 - 4x

y ( x + 3)  = 6  - 2x                z( x + 3)  = 30 - 4x

y =  (6 - 2x) / ( x + 3)             z =  ( 30 - 4x) / ( x + 3) 

yz  =  6 - 4y - 2z

(6 - 2x) /( x + 3)  *  (  30 - 4x)/ (x + 3)   =   6   - 4 (6 - 2x)/(x + 3 )  - 2 ( 30-4x)/ ( x + 3)

Multiply  through  by (x + 3)^2

(6- 2x)  * (30 - 4x)  =  6 ( x + 3)^2   - 4 (6-2x)(x + 3)  - 2(30-4x)(x + 3)   

180 - 84x + 8x^2   =  6(x^2 + 6x + 9)  - 4(6x - 2x^2 + 18 - 6x)  - 2 ( 30x - 4x^2 + 90 - 12x)

180 - 84x  + 8x^2  =  6x^2 + 36x + 54  - 24x + 8x^2 - 72 + 24x  - 60x + 8x^2 - 180 + 24x

Simplify

180 - 84x  + 8x^2  = 22x^2   -198

14x^2  + 84x - 378  =  0           divide through by 14

x^2  + 6x  -  27  =  0      factor

(x - 3) ( x + 9)  =  0

The positive solution is   x  -3   = 0    ⇔   x  =  3

thanks

EP is off-line; I'll try to explain it in a way that makes sense to me.

To find the four fourth roots of -4:  z  =  (-4)^1/4

I'm assuming that you are familiar with  r·cis( theta ) form; if not, this may not make much sense.

If you graph z = -4 on the complex plane, you'll notic that its distance from the origin is 4;

this means that  r  =  4

You'll also notice that it's directly west; therefore, its angle is  180^o  (if you wish, you can change this into radians).

So, the complex number  -4 =  -4 + 0i  =  4cis(180^o)

To find the primary fourth root, find the fourth root of r and divide the angle by 4.

So, the primary fourth root of -4  is  4^1/4cis( 180^o/4 )  =  sqrt(2)·cis( 45^o )

To find the other fourth roots, divide 360^o by 4 to get 90^o and add this amount onto the angle

of the primary fourt root three times:

So, your four fourth roots are:

sqrt(2)·cis( 45^o )

sqrt(2)·cis( 45^o + 90^o )  =  sqrt(2)·cis( 135^o )

sqrt(2)·cis( 135^o  + 90^o )  =  sqrt(2)·cis( 225^o )

sqrt(2)·cis( 225^o + 90^o )  =  sqrt(2)·cis( 315^o )

sqrt(2)·cis( 45^o )  =  sqrt(2)·cos( 45^o ) + i·sqrt(2)·sin( 45^o )  =  1 + 1i

sqrt(2)·cis( 135^o )  =  -1 + i

sqrt(2)·cis( 225^o )  =  -1 - i

sqrt(2)·cis( 315^o )  =  1 - i

3.14/1.41 = about 2.22

ahh ok thanks!

You can't have a 1 on the first die:        5/6

You can't have a 1 on the second die:  5/6

You can't have a 1 on the third die:       5/6

Answer  =  5/6 x 5/6 x 5/6  =  125/216.

I think that should be    9/2    rather than 11/2

like this:

pi * ( 10/2)^2 * 200  -  pi * (9/2)^2 * 200 

z^2 = 2i

z = +- sqrt 2i =   1+i      or  1 -i    or   -1 -i     or   -1 + i

like the 11/2 part

Hey CPhill!
I don't get the part where
 

pi * ( 11/2)^2 * 200  -  pi * (10/2)^2 * 200  =

can u please explain thanks.

The  volume  of  the pipe  is just  the  difference  in the volumes of two cylinders  =

Volume  of "outside" cylinder  -  Volume of "inside" cylinder  =

{2m  =  200cm)

pi * ( 11/2)^2 * 200  -  pi * (10/2)^2 * 200  =

200pi [ 5.5^2 - 5^2  ] = 

200pi  * 5.25  =

1050 pi   cm^3

So....the weight  is

1050 pi cm^3  *  (8.2)g / cm^3    =    [8610 pi]   g  ≈  27049 g

The circumference of a circle is pi^2.  Find the length of a side of a square inscribed in the circle. 

Circumference is π • d.  Since this circumference is π • π that means d = π 

The diameter, i.e., π, is the diagonal of the inscribed square.  

Therefore, a side of the square is π / sqrt(2).  In numeric value approx  3.1416 / 1.414  =  about 2.22 

_.

I only see the first answer from the choices given to me, I don't see the rest

a)   The  center  of  the circle  is   (2, -3)

Using  the  formula  for the distance from a point to a line, we have

l  4(2)  - 5(-3)  - 20  l             l  3 l              3

___________________  = ______  =   ________ 

   sqrt  (4^2 + 5^2)               sqrt 41        sqrt (41)

b)  greatest  distance  from a point  on the circle to  the  line   =

distance found in (a)   +    radius   =

3 / sqrt (41)  +  2   

One where is the diagram

Two please kindly fix your latex.

Thank you.

Make a 6 x 6 grid.  In each square put the possible combination.  Count the ones that satisfy the conditions. 

It'll look something like the one below

1,1   1,2   1,3   1,4   1,5   1,6  

2,1   2,2   2,3   2,4   2,5   2,6  

3,1   3,2   3,3   3,4   3,5   3,6  

4,1   4,2   4,3   4,4   4,5   4,6  

5,1   5,2   5,3   5,4   5,5   5,6  

6,1   6,2   6,3   6,4   6,5   6,6

You see in the first two rows that none of the pairs satisfies the requirement that a be divisible by 3 

In the third row, 3,3 and 3,6 work.  In the sixth row, 6,3 and 6,6 work. 

That's four of them.  So, 4 out of 36 is 4/36 and reduces to 1/9.  You were right.  Good work! 

_.

r - rcos  θ   =   6

r   - r ( x/r)   =  6

r  -  x   =  6

r   =   x +  6           square  both sides

r^2  =  x^2  + 12x  + 36

x^2 + y^2  =  x^2  + 12x  +  36

y^2  =  12x  +  36

12x  =  y^2  -  36        divide through by  12

x  =  (1/12)y^2  -  3

1.Find the area of the convex quadrilateral with vertices (1, 5), (2, 3), (7, 6) and (7, 1).

See the image here

We can  use  something known as Pick's Theorem  to solve this

Note...a lattice  point  =  a  point with integer coordinates

Area =  number of boundary lattice points /2  +  number of interior lattice points  - 1   =

   (4/2)    +   16  -  1   =

2   +   16   -    1     =

  17

Question #1:  On graph paper:

--  plot point A(1,5)

--  plot point B(7,6)

--  plot point C(7,1)

--  plot point D(2,3)

Connect sides AB, BC, CD, and DA to get the quadrilateral ABCD.

Now, find point E(7,5) and point F(7,3).

Draw a line segment from A to E and another line segment from D to F.

Quadrilateral ABCD is now divided into a top triangle (ABE), a bottom triangle (DFC)

and a middle trapezoid.

If we find the areas of these three figures and add them together, we will have the area

of quadrilateral ABCD.

To get the area of the top triangle (ABE):

-- the base is AE (find this length)

-- the height is EB (find this length)

-- and use the formula:  A  =  ½ · base · height  =  .........

To get the area of the bottom triangle (DFC)

-- the base is DF (find this length)

-- the height is FC (find this length

-- and use the formula:  A  =  ½ · base · height  =  .........

To get the area of the trapezoid (AEFD)

-- one of the bases is FD

-- the other base is EA

-- the height is EF

-- and use this formula:  A  =  ½ · h · (base₁ + base₂)  =  .........

Now, add these areas together to get the total area ....