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The cost of the gallon is $4.05, it just didn't come up earlier cuz of latex

The smallest positive integer t is 13.

W/9 = -180    Multiply both sides of the equation by 9

w= 9 * -180 = - 1620

Boone =x

spot= x + 45

lex = x-32

summed, they equal 199

x      +     x+45      +        x-32   =  199

3x = 186

x = 62= Boone      Spot = 62+45      Lex =   62-32

Sqrt (8^2 - 6^2) = sqrt(28)= 2 sqrt7

sqrt(8^2 - 6^2) = 4 √ 3.

The distance between the horizontal lines is 1

The tangent of angles ACD and BCD is ⁷/₁₈   or  0.38888889  ==>  t

a = 1 * t = 0.38888888

b = 2 * t = 0.77777777

c = 3 * t = 1.16666666

d = 4 * t = 1.55555555  

e = 5 * t = 1.94444444

f = 6 * t = 2.33333333

g = 7 * t = 2.72222222

h = 8 * t = 3.111111111

i = 9 * t  = 3.50000000

Add them up and you'll get 17.5 

We might also say that AB = 8 sqrt (13), and as OD=8, BD = sqrt(OB^2 - OD^2) = sqrt(208 - 64) = 12.

Extend BD to meet the circle at C. Then, BC = 2 BD = 24. Then, 832 -576 = 256, so AC = 16. Thus, AD = sqrt(AC^2 + DC^2) = sqrt(256+144) = sqrt(400) = 20.

Please do 2398 divided by 96 and you should get your answer! If you still need help or want me to check your answer just tell me!

- Grace

I can tell you that $cos(x) + sin(x) = \frac{5}{\sqrt{13}},$ so $2 \cdot cos(x) + 2\cdot sin(x) = \frac{10}{\sqrt{13}}.$ Subtracting, $sin(x) = \frac{3\sqrt{13}}{13}.$ Can you find $cos(x)$ now?

Can you solve it another way? Or can you continue with the work I have? I haven't learned about tan and stuff yet so...

You are off to a good start. We know that because BDI is a 3-4-5 triangle, it is right, and because an angle bisector is a perpendicular in triangle ABC, ABC is isosceles. As ABC is isosceles, a perpendicular from A is a median of BC! Therefore, BC = 2*BD = 8. 

I'm going to coordinate bash. Let B be (-4,0) and C be (4,0). Then, reverse-using the formula for the incenter's coordinates, or (0,3), A = (0, 96/7).

We have that the height is 96/7, so the area is 96/7 * 4 = 54.8571428571 or 54 6/7.

jugoslav, nice to see we agree :)

z = a + bi               z conjugate =    a - bi

So

2 ( a + bi)  +  3  ( a - bi)  =  -25 - 8i

2a  + 2bi  +3a  - 3bi =   -25  - 8i

(2a + 3a)  = - 25             and ( 2b - 3b)i =  -8i

5a  =  -25                              -bi = -8i

 a  = -5                                    b  = 8

z  =  -5  + 8i

In a triangle, ABC,  angle bisectors  AD, CF and BE meet at a point I. If DI=3, BD=4, and BI=5, then compute the area of triangle ABC.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

∠B = 2(tan^-1(3/4)) = 73.73979529º

AD = tan∠B * 4 = 13.71428571

Area = BD * AD = 54.857142286 u²

3 ⇒ 8

8 ⇒ 1

1 ⇒ 6

6 ⇒ 11

11 ⇒ 4

4 ⇒ 9

9 ⇒ 2

To  have real roots,  the  discriminant  must  be  >=  to  0

Therefore

(2p+1)^2   - 4 ( 3/4) (p^2 + 2)   >= 0

4p^2  + 4p  + 1   -   3 (p^2 + 2)  > =   0

4p^2  + 4p  + 1   - 3p^2  -  6    >= 0

p^2 + 4p  -  5   >=  0        factor

(p + 5) ( p -1)  > =  0

Note  that this will be  true when   p  =  [ 1, inf)    or  when  p =   ( - inf,  -5  ]

p  = 1  is the  minimum  positive  value that will produce  real roots 

If we join the  ceters of these  circles, we have an equilateral tirangle with  sides of 10

We can compite  the  distance, D, from the  center of the large circle to the center of any of the small circle as

10^2  =  2D^2   - 2D^2 cos (120°)

100  =  2D^2  - 2D^2 (-1/2)

100  =  3D^2

100/3  = D^2  

D = 10/sqrt (3) 

So....the radius of the larger circle is   10/sqrt (3)  +  5   ≈  10.774    

i =   7/2  *  (9/9) =  3.5

h =    (7/2)  * (8/9)  

g  = (7/2) * ( 7/9)

f = (7/2) * (6/9)

e = (7/2)* (5/9)

d = (7/2) * ( 4/9)

c = (7/2) * (3/9)

b =(7/2) * (2/9)

a = (7/2) * (1/9)

Their sum  =   

(7/2)  ( 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1) / 9   =

(7/18)  *  ( 45)  = 

(45/18)  * 7   =

(5/2) * 7  = 

35 / 2  =  

17.5

Let  the  portion of the wall that   Bill  can  paint in  one hour  be  1/x  =  his rate  where x is the  number  of  hours that Bill takes to  paint the wall by himself

Let  the portion of the wall that  Peter can paint in one hour  = 1/ ( x + 2)   = his rate  where x + 2  is the  number of hours that Peter  takes to paint the wall by himself

And

Bill's rate * 7hours  +  Peter's rate * 7 hours =   1  whole job done.....so...

1/x * 7   + 1/(x + 2) * 7  = 1

7/x  +  7/(x + 2) =  1

7 ( x + 2)    +  7x

______________   =     1

     x ( x + 2)

14x  + 14  =   x ( x + 2)

14x + 14 = x^2  + 2x    rearrange as

x^2  - 12x  -  14 =   0        complete the square on x

x^2  -12x  +  36   =   14  +   36

(x - 6)^2 =    50        take the positive root

x - 6 =  sqrt (50)

x  =    sqrt (50 ) +  6

x +  2 =       sqrt (50 )  +  8   ≈  15.07  hrs =  Peter's time  =  15  +  .07 *60   ≈  15 hrs  4 min

Very nice, JoLink....!!!!

Let O = (0, 0), D = (0, -8). Then B = (-12, -8) because OD = 8 and OB = 4sqrt(13), so by the Pythagorean Theorem, BD = 12

Then, since AB is a line segment passing through O, we have that A = (12, 8). Use the distance formula from (0, -8) to (12, 8) and get sqrt((12-0)^2+(8-(-8))^2) = sqrt(12^2+16^2) = 20, so AD = 20.
 

This is probably not the most efficient way, but We Still get the right answer.

The height of this trapezoid is AB and the bases are ZD and WC

We have

120 = (1/2) AB  (ZD + WC)

120 = (1/2) 12 ( ZD + 6)     multiply through by 2

240 = 12 (ZD + 6)      divide both sides by 12

20 =  ZD + 6       subtract 6 from both sides

14 = ZD

Now   angle DBW = angle BDZ

And angle ZQD = angle BQW

And since AZ = WC

Then  ZD = BW

So by AAS....triangle ZQD is congruent to triangle WQZ

And the altitude of trangle DQZ    = altitude of triangle BQW

And since twice these altitudes = AB....then each altitude = 1/2(AB) = 6

So....the area of triangle BQW   =

(1/2) BW * altitude of BQW   = 

(1/2) (14)(6) =

(1/2) (84) =

42 units^2

TYTYTY :D

Either  

2(2d - 40)  + (d + 20)  =  180   simplify

4d  -  80  +  d  +  20  =  180

5d  - 60  =  180

5d   =   240

d  = 240/5 =  48°

Or

2(d + 20)  +  (2d - 40)  = 180

2d + 40  + 2d    - 40    =  180

4d   =   180

d  = 180/4  =  45°

The numerator is defined for all x

The  x values that  make the  denominator  =  0  will  not be defined

x^2  -20x  +  100    =    0       factor

(x - 10) ( x - 10)   =  0

Setting  either  factor to 0  and solving for  x

x  -  10   =  0

x = 10   =  the  value of x that makes  the rational  function undefined