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1.     Remember  that  i^2  = -1       i^3   =  -i        i^4    =1

P(1)  =   (1)^3  + (1 - i)(1)^2  + 2i(1)  + 3 +4i

                =     1   + 1   - i  + 2i  +  3 + 4i

                =      5  +  5i

P(i)  = (i)^3 + (1  -i)(i)^2  + 2i(i)  + 3 + 4i

        =  -i   + (1 - i)(-1)  + 2i^2  + 3 + 4i

        =  -i  - 1 + i  + 2(-1) +  3  +  4i

        =     -1   -   2  + 3   +  4i

        =    4i

P(1 + i)  =  (1 + i)^3  + (1 - i) (1 + i)^2  + 2i (1 + i)  +  3  + 4i

                Note  that we can write

                 ( 1 + i)  [ (1 + i)^2  +  (1 - i) ( (1 + i)  + 2i ]  +3 + 4i     =

                 (1 + i)  [ (1 + 2i +i^2  + 1  - i^2  + 2i   ] +  3 + 4i  =

                 (1 + i)  [ 1  + 2i  - 1  + 1  + 1  + 2i ]  + 3  + 4i   =

                 (1 + i)  [ 2 + 4i)  + 3 + 4i =

                  (2  + 2i + 4i + 4i^2) + 3 + 4i  =

                  (2  + 6i - 4 ) + 3 + 4i  =

                   (6i - 2) + 3 + 4i  =

                    1 + 10i

P(1 - i)  =  (1 - i)^3  + (1 - i)(1 - i)^2  + 2i (1 - i) + 3  + 4i

                And we can write

                (1 - i)  [ (1 - i)^2  +(1 - i)^2  + 2i ]  + 3  + 4i  =

                ( 1 - i)  [ 1  - 2i + i^2  + 1 - 2i  + i^2  + 2i ] + 3 + 4i   =

                (1 - i)  [ [ 1  - 2i  - 1  + 1  -2i  -1  + 2i ] + 3 + 4i   =

                (1 - i) [ -2i]  + 3 + 4i    =

                 -2i  + 2i^2  + 3  + 4i  =

                  -2i  - 2 + 3  + 4i   =

                   1  +  2i

The hypotenuse of a right triangle measures 2√15 centimeters and its shorter leg measures 2√6  centimeters. What is the measure of the larger acute angle of the triangle? Round your answer to the nearest tenth of a degree.

Die Hypotenuse eines rechtwinkligen Dreiecks misst 2 \(\sqrt{15}\) Zentimeter und die kürzere Kathete 2 \(\sqrt{6}\)Zentimeter. Was ist das Maß für den größeren spitzen Winkel des Dreiecks? Runden Sie Ihre Antwort auf das nächste Zehntel eines Grades.

Hello valval2205!

A right triangle ABC

\(c=2\cdot \sqrt{15}\ cm\\ b=2\cdot \sqrt{6}\ cm\)

\(\alpha=arccos(\frac{b}{c})=arccos(\frac{\sqrt{6}}{\sqrt{15}})\)

\(\alpha=50.8^o\)

  !

THANK YOU CPhill!!!!! (WOW! YOU ARE SO SMART!)

Just type   " ! "   after your number and enter......the calculator recognizes this symbol

It is the shift key and 1.

Let  A  =  A

Let B =  A + d

Let C  = A  + 2d

Note that    A  +  C  will be supplementary......so....

A + (A + 2d)  = 180

2A  +  2d  = 180         divide through by  2  

A  +  d    =   90

And     B   + D   are also supplementary

So

(A + d )  +  D  =   180

90  + D   =  180     subtract  90  from both sides

D  =  90°

Note  that  B  must also   = 90°

And   A , B , C   are in arithmetic progression  if  A  = 60°  and  C  =120°

Consider x to be the number of days

350,000 + 200x = 1,000,000

200x = 650,000

2x = 6,500

x = 3250 days

I assume you mean factorial... the symbol is "!"

n! = n! * (n - 1)! * (n-2)! ... * 1

For example

5! = 5 * 4 * 3 * 2 * 1 = 120 

Another way to do this problem:

x + 125  =  a²                                                        x + 176  =  b²

x + 176  =  b²                                                       x + 125  =  a²

Subtracting the first equation from the second:           51  =  b² - a²

                                                           factoring:        17·3  =  (b + a)(b - a)

This means that  b = 10  and  a = 7

So:  x + 125  =  a²      --->     x + 125  =  7²     --->     x + 125  =  49     --->     x  =  -76

       x + 176  =  b²     --->     x + 176  =  10²    --->     x + 176  =  100   --->     x  =  -76

The  largest angle  =  (4) / ( 2 + 3 + 4)  * 180   =  (4/9) * 180  =  4 * 180/9   = 4 * 20  = 80° 

a^2 - b^2  =  ([176+ n] - [125 +n])

a^2  - b^2  =  176 - 125

a^2 - b^2  =  51

Note  that   if   a  =10   and b = 7  this is true  because     10^2  - 7^2  =  100 - 49  - 51

So  a^2 =  100     and  b^2 =  49

So

100 - 176   =  n  =   -76

49  - 125  = = n  = -76

So....the  number  is   -76

A cube 3" on each edge, is painted entirely red. It is then cut apart into 1" cubes. How many of these will have paint on exactly two faces 

I'm visualizing that in my mind, and the answer is 12 cubes.  The middle one on each of the sharp edges. 

The 6 in the center of the faces will have one side red. 

The 8 corners will have three sides red. 

The 1 buried in the center will have no red paint. 

Add them all up (12+6+8+1) and that accounts for all 27 cubes. 

_.

All five digits are different



The digit in the ten thousands place is greater than 7         so it is 8 or 9 

The sum of all five digits is 18,                                     others add to 10 or 9                                 

The greatest digit is equal to the sum of the other four digits.     So the the biggest is 9 and the others add to 9

None of the five digits are 1                                                0+2+3+4=9

so the digits from biggest to smallest are   9,4,3,2,0

The biggest possible number is   94 320    just like guest said.

Let  the rate of the plane in still air  = P

Let  the rate of the wind   =  W

Distance  / Rate  = Time  (in hours).....so.....

8260                                                         6560                                              

______  =     7                      and          ______  =    8

R + W                                                      R  - W

8260  / 7  =  R + W                              6560  / 8   =  R - W

R + W  =  1180    (1)                                R  -  W   = 820    (2)

Add (1)  + (2)  and we get that

2R   =  2000    divide both sides by  2

R  =  1000    (mph)

And

R - W  = 820

1000  -  820  =  W

180  (mph) =  W

Agree with  geno

We will have  two equations but   3 unknowns....so....we could have infinite solutions

If we knew the volume.....we could solve  this

Let   x  be  the amount  invested at 6%    and (20000 - x)  = the amount invested at 7%

We have  this equation  { Principal * Rate  = Interest }

.06x  + .07 ( 20000  - x)  =  1280    simplify

.06x  +  1400   - .07 x   =  1280

-.01x  +  1400   = 1280      subtract  1400 from both sides

-.01x  = -120      divide both sides by  -.01

x  =  $12000  =   amt  at 6%

20000 - x =  $ 8000   = amt at 7%

The slope  of  this  line   =   

[ 1 - - 5  ]  /  [ 6  - - 3 ] =  6 / 9  =  2 / 3

So...using either point.......the equation of the  given line is

y  = ( 2/3) ( x - 6 )  +  1     multiply through by  3

3y  = 2 ( x - 6)  +  3

3y  = 2x  - 12  +  3

3y  =  2x  -  9      rearrange  as

2x  - 3y      =  9

A =  2       B   = -3     C   = 9

A  +  B  +   C    =    8

Since ABC is isosceles.....then angle  ABC  = 45°

PQ   is parallel to   BA

Therefore...angle QPC   = angle  ABC  =45°

And angle  MPQ   =  60°

So angle  MPB   =180 - 45 -60  =  75°

Using the Law of Sines....we can  find  MP  = the side of  MPQ  thusly

MB  / sin MPB   =  MP /sin ABC

36   / sin (75)  = MP / sin 45      

36 sin 45  /   [(sin (30 + 45)    = MP

36 sin 45  / [ sin 45cos 30 + cos 45 sin 30]                     {sin 45  = cos 45 }

36sin 45  /  [ sin 45  *  ( √3/2   + 1/2)  ] =

2 * 36 /[    √3  + 1 ]  =    

72 / [ √3 + 1]  =

72 [ √3  -1]   /  [ 2 ]   =

36  [√3  -1  ]   =  MP

So.....the area  of triangle  MPQ   =

(1/2 ) (MP*^2 * sin (60)  =

(1/2) [ 36 * [√3 - 1] ]^2  *  (√3  / 2 )   ≈

300.74 units^2   

Just as Dragan found  !!!!!

a=1;b=0;c=0;d=0;e=0;f=0;p=0; cycle:n=a*100000+b*10000+c*1000+d*100+e*10+f; if(a>b and b>c and c>d and d>e and e>f, goto loop, goto next);loop:printn," ",;p=p+1; next:f++;if(f<10, goto cycle, 0);f=0;e++;if(e<10, goto cycle, 0);f=0;e=0;d++;if(d<10, goto cycle,0);f=0;e=0;d=0;c++;if(c<10, goto cycle,0);f=0;e=0;d=0;c=0;b++;if(b<10, goto cycle,0);f=0;e=0;d=0;c=0;b=0;a++;if(a<10, goto cycle,0);print"Total = ",p

OUTPUT = 210 such numbers

a=20;i=1;c=(a^3 - a); if(c%12==0, goto4, goto6);printi,"-",a;i=i+1; a++;if(a<100, goto2, 0);print i
1 - 20
2 - 21
3 - 23
4 - 24
5 - 25
6 - 27
7 - 28
8 - 29
9 - 31
10 - 32
11 - 33
12 - 35
13 - 36
14 - 37
15 - 39
16 - 40
17 - 41
18 - 43
19 - 44
20 - 45
21 - 47
22 - 48
23 - 49
24 - 51
25 - 52
26 - 53
27 - 55
28 - 56
29 - 57
30 - 59
31 - 60
32 - 61
33 - 63
34 - 64
35 - 65
36 - 67
37 - 68
38 - 69
39 - 71
40 - 72
41 - 73
42 - 75
43 - 76
44 - 77
45 - 79
46 - 80
47 - 81
48 - 83
49 - 84
50 - 85
51 - 87
52 - 88
53 - 89
54 - 91
55 - 92
56 - 93
57 - 95
58 - 96
59 - 97
60 - 99
---------------

All the above 60 integers when expressed as: [C^3 - C] % 12=0
99 - 20 + 1=80
60 - 1 + 1 =60
The probability is: 60 / 80 =3 / 4

https://web2.0calc.com/questions/graph-question_4#r1

x + 5y  =7 

15x + Ay   = B

The slope of the first line =  -(1/5)

So..since the lines are perpendicular  ..the slope of the second line must  be    =  5

This implies that     -15/A = 5     ⇒  A  = -3

If the lines  intersect  at   (-8, 3)......the  for the second line, 

 15(-8) + -3(3)  = B

-120  - 9  = B

B  = -129

So the equation of the second line must   be

15x  - 3y  = -129

So   (A, B)  =  (-3 -129)

Here's a graph  : https://www.desmos.com/calculator/zyizjg9kkt

In a rectangular prism, you have three variables:  length, width, and height.

But, you have only two formulas: one for total length of the edges, the other for the total surface area.

In my opinion, this can't be solved.

2 second blast + 8 second pause= 1 blast every ten seconds

60 seconds in a minute= 6 blasts a minute

60 minutes in an hour = 360 blasts an hour

Then just multiply that by 3.5 and that is your answer

Question #1:  This will be a Fibonacci Sequence:  1, 1, 2, 3, 5, 8, 13, 21, 34, ...

To get the 7^th term to be 6, multiply each term by 6/13, giving:

     6/13, 6/13, 12/13, 18/13, 30/13, 78/13, 126/13, 204/13  =  480/13

Question #2:  1,2,-3, 4,5,-6, 7,8,-9, 10,11-12, ...

Adding each set of 3 numbers give this sum:  0, 3, 6, 9, ...

This becomes an arithmetic sequence; to get the sum of the (original) 24 numbers:

    Sum  =  8(0 + 21)/2  =  84

Then, you must add the 25^th number:  84 + 25  =  109

Question #3:  The sequence 1, 5, 12, 22, 35, ...  Is the pentagonal numbers.

A formula to find the n^th number is:  P_n =  n(3n - 1)/2

So the 10^th number is:  10(3·10 - 1)/2  =  145