All Answers 358090 Answers

(f ° g)x = (g ° f)x

2(-3x + c) + 1  =  -3(2x + 1) + c

-6x + 2c + 1  =  -6x - 3 + c

2c + 1 =  - 3 + c

 c = -3 - 1

c = -4

Number of square ft / [ number of people * time ]   =  sq ft that one person can paint in  one  minute

3600 / [ 50 * 20 ]   = 4000 / [ 25 * x ]

3600 [ 25 *x ] = 4000 [50 * 20 ]

90000 x = 4000000

9x =  400

x =   400/ 9  =   44.4 min

[A ] * k^2  =  [B]

k^2  = [B ]  / [ A]

k^2  =  9 / 16

k = 3 / 4

a(-3)^2  - (3)b + c =  5

a(-1)^2  - 1b + c = 7

a(2)^2 + 2b + c  = -1

9a  - 3b + c =  5     (1)

1a  -1b + c   = 7  →  -1a + 1b - c  = -7     (2)

4a  + 2b + c  = -1    (3)

Add (1),(2)

8a -2b = -2    →  4a - b = -1 →  b = 4a + 1    (3)

Add (2) , (3)

3a + 3b = -8       (4)

Sub (3)  into 4

3a + 3(4a + 1) = -8

15a + 3   = - 8

15a  = -11

a = -11/15

b = 4(-11/15)+ 1 =  -29/15

a - b + c   =  7

(-11/15) - (-29/15) + c  =7

18/15 + c  = 7

c = 105/15 - 18/15 = 87/15   = 29/5

a* b * c   =   (-11/15)(-29/15)(29/5)   =    9251  /  1125

OB / OA =  4

60  = 4 OA

60  / 4 =  OA  =   15

abs (x)   = sqrt (x^2)

sqrt [ (x^2 -30x -1)^2 ]  + [sqrt [ (x^2 -30x + 29)^2 ] =  30        square both sides

(x^2 - 30x - 1)^2  + 2 sqrt [ (x^2 -30x -1)^2 (x^2 -30x + 29)^2] +(x^2 -30x + 29)^2 =  900

Solving this wouldn't be easy

Using a little technology  (WolframAlpha)  shows the solutions to be

[  -1 / (15 + sqrt (226)) , 1 ]   U  [ 29, 15 + sqrt (226 ] 

Using the square of the distance formula

(a-7)^2 + ( b - 1)^2  = (a -2)^2 + (b - 0)^2

a^2 - 14a + 49 + b^2 - 2b + 1  =a^2 - 4a + 4 + b^2

-14a -2b + 50  = -4a + 4

-10a + 46  = 2b

b = 23 - 5a            

Again

(a  - 3)^2  + (b - 5)^2  = (a - 2)^2  + (b - 0)^2              {sub in 23 - 5a  for b }

a^2 - 6a + 9 + ( 18 - 5a)^2  = ( a - 2)^2 + (23 - 5a)^2

a^2 - 6a + 9 + 324 - 180a  + 25a^2  = a^2 - 4a + 4 +25a^2 - 230a + 529

-6a + 9 + 324 - 180a = -4a + 4 -230a + 529

-186a + 333  = -234a + 533

48a  = 200

a = 200 / 48   =  25/6  

b = 23 - (5)(25/6)  =  23  - 125/6 = 13/6

(a,b)  = (25/6 , 13/6)

A     3    E    4    B

                F             

C                      D

[EAD] = (1/2) (3/7)AB * AC

9 = (3/14)AB * 10

9 = (30/14)AB

9 = (15/7)AB

9 ( 7/15)  = AB

(63/15) = AB  =   21/5  

[ABCD ]  =  AB * AC  =  (21/5) (10)  =  42

(-4,0) and (0,2)  are on the  graph

The slope is  [ 2 -0 ]/ [ 0  - -4 ] =  2/4 = 1/2

Using the slope and (0,2)  the equation of the line is

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

2y = x  + 4

Rearrange as

x - 2y =  -4

Angle BDC   = 30

To find base CD

CD / sin DBC  = BC / sin BDC

CD / sin 80  = 6 / sin 30

CD  = 6 * sin 80  / sin 30  =  6 * sin 80 / (1/2)  =  12sin(80)

Area of triangle BDC =

(1/2) BC * CD * sin 70  =

(1/2) 6 * 12sin 80 * sin70 =

36sin 80 * sin 70  ≈  33.3

To find BD

BD /sin70 = BC / sin30

BD /sin70 = 6 / (1/2)

BD / sin70 = 12

BD = 12sin70

Area of triangle ABD

(1/2) AB *BD sin ABD =

(1/2) *3 * 12sin (70)*sin (30)  =

18 sin (70) * (1/2) =

9 sin (70)  ≈ 8.46

Area of trapezoid  =  Area of triangle BDC + Area of triangle ABD  = 

33.3  + 8.46 ≈  41.76

1)

To simplify this equation, we want to eliminate some of the terms. Set 

\(x=a_1-a_2\)

\(y=a_2-a_3\)

Therefore:

\(a_3-a_1=-(a_1-a_3)=-(x+y)=-x-y\).

Now our equation is:

\(|x|+2|y|+3|-x-y|=1\). 

\(|x|+2|y|+3|x+y|=1\).

One way to approach this is by graphing. 

We split this into 3 sections:

One for \(x \ge 0, x < 0\)

\(y\ge0, y<0\)

For \(|x+y|\) we have

\(x>-y, x < -y\).

Here is a graph, with \(x=0, y=0, x=-y\) labeled. (The green one is \(|x|+2|y|+3|x+y|=1\)).

We see the largest possible value of \(|x|\) in this graph is \(\frac{1}{3}\).

Therefore, the maximum value is \(|a_1-a_2|=\frac{1}{3}\).

60pi = pi * r^2 * (83/360)

60 (360/ 83)  = r^2

r  = sqrt [ 60 * 360  / 83  ] ≈  16.13

Not enough info to solve this....

\(\lfloor t\rfloor = 3t + 4 - \lfloor 2t \rfloor \)

See the graph here 

blue = floor (t)

green  = 3t + 4 - floor (2t)

Note that the graphs never intersect so there are no  solutions for  t

A period is the distance between each repetition.
It's easy to see that after 12, f(m) starts repeating again.

Therefore the period is 12.

The vertical shift of this function is how much it's shifted up.

The sine function started at 50, so the vertical shift is 50 (also it's easy to see 50 is the midpoint).

Because 3 is right in between 0 and 6, it is the max value, similarly 9 is the min value.

So 50-33 = 17, and 67-50 = 17 too, so the amplitude is 17.

Therefore, The amplitude of the sine function is 17. The period of the sine function is 12. The vertical shift of the function is 50.

If it interests you, the equation of the function is \(f(m)=-17\sin(\frac{2\pi}{12}m)+50\).

The function inside, \(f(x) = \left\lfloor\frac{2 - 3x}{3x + 1}\right\rfloor\) is \(\frac{2-3x}{3x+1}\).

Using what we know about rational functions, set the inside function,  to g(x), \(g(x) = \frac{2-3x}{3x+1}\) has a x intercept of 2/3, and a horizontal asymptote of -1, and a vertical asymptote at x = -1/3.

Using this information, we can determine the relative shape of the graph.

Because the function has an x intercept at 2/3, and the function only goes down after that, but will not go down past -1, since that is our horizontal asymptote. Plugging in values from \(1\le x \le \infty\) will always result in a output of \(-1\le g(x) \le 0\).

Therefore because \(f(x)=\lfloor\frac{2-3x}{3x+1}\rfloor\), if x is from 1 to 1000 will always result in -1.

Therefore \(f(1)+f(2)+f(3) + \dots + f(999) + f(1000) = -1000\).

x^2 - 4x + y^2 - 12y = 0

x^2 - 4x + 4  + y^2 - 12y + 36 =  40

(x - 2)^2  + (y - 6)^2  =  40

This is  a circle  centered at ( 2,6)   with a radius of sqrt (40)

The max lies on the intersection of the line y = x + 4   and the  circle

x + y   max  =    [2 + sqrt (40)cos (45°)]  + [6 +  sqrt (40)sin(45°)]  =

2 + 2sqrt (10) * (1/sqrt 2)   +  6 + 2sqrt (10) * (1/sqrt 2)  =

[2 + 2sqrt (5)]  +    [6 + 2sqrt (5)]  =

8 + 4sqrt (5)  =

4 (2 + sqrt (5) )

Slope  =  [ b^2 - a^2 ] / [ b - a ] =  [ (b + a) (a - b) ] / [(a - b)] =  2

So

a + b =  2

a^2 / b  +  b^2 / a  =     [ a^3 + b^3 ] / [ab ] ^2  =   [(a + b)  (a^2  + b^2  - ab ) ] /  [ab]

x^2 -25x + 80  = -28x +  75       simplify

x^2 + 3x + 5 =  0

We have the form  mx^2 + nx + c =  0

Sum of roots

a + b =  -n/m =  -3/1 =  -3

Product of  roots  =  ab =  c / m =  5/1 =  5

a + b =   -3       square both sides

a^2 + 2ab + b^2 =  9

a^2  +2(5) + b^2  = 9

a^2 + 10 + b^2 = 9

a^2 + b^2  =  -1

So

[(a + b)  (a^2 + b^2 - ab) ] /  [ab]  =

[ ( -3)( -1 - 5) ]   / [5] =

[ 18 ] / [5 ]   =  

18 / 5 =  3.6

Let G be the total  number of  games   you  play

Let .75 G be the number of  games  you  have played

Let . 35 (.75)G be the number of  games  you  have won

Let .25G be the games  left

Let  .50G = the number of  wins  you  must have

And number of wins you  need - Number of  wins  you  have =  No. of  wins you need in  your remaining  schedule

So

[ No. of wins you  need - No, of  wins you have  ] / [ No. of Games Left ]

[ .50G - .35 (.75)G ] /  .25G  =

(.2375 G )  / ( .25G) = .95  = 95%    (you need to  win  95%  of your remaining  games )

x + y + 8z  =  10

x + 9y -4z  =  7

kx + 4y +2z = 3

Multiply the second equation  by 2

2x + 18y  - 8z  = 14

add to the first equation

3x + 19y  = 24        (1)

Multiply the last equation by 2

2kx + 8y + 4z  = 6

add to the second equation

(2k + 1)x + 17y = 13    (2)

Multiply (1)    by 17    and (2) by 19

51x + 323y  = 408

(38k + 19)x + 323y =  247

This system will  have no solution  when

38k + 19 = 51

38k  = 32

k = 32 / 38  =  16  /19