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It is given that m || n , m∠1 = 65° , m∠2 = 60° , and BD bisects ∠ABC .

Because of the triangle sum theorem,  m∠3 = 55°.

By the definition of bisector, m∠ABC = 110°. (unsure of this one, but no other choice seems right)

m∠5 = 110° because vertical angles are congruent.

Because of the same-side interior angles theorem, m∠5 + m∠6 = 180°.

Substituting gives 110° + m∠6 = 180° .

So, by the subtraction property of equality, m∠6 = 70°

oof i am trying to do it but right of the bat i am stuck with an ugly looking \(165-\frac{-21}{288}\)

and some how i am supposed to do this five times, each time it gets more ugly...

midpoint   =   ( [-3 + 2]/2 ,  [-5 + 7]/2 )   =   ( -1/2 ,  2/2 )   =   ( -0.5 ,  1 )

the  x  coordinate of the minimum   =   -(-15) / [ 2(3) ]   =   15/6   =   5/2

the  y  coordinate of the minimum   =   3(5/2)^2 - 15(5/2) + k

1   =   3(5/2)^2 - 15(5/2) + k

1   =   3(25/4) - 75/2 + k

1   =   75/4 - 75/2 + k

1  =   -75/4 + k

1 + 75/4   =   k

79/4   =   k

Sure !!.

bleh i will anyway

ok sure! can i use pen and paper?

on each dice there are 2 square numbers (1 and 4). so for each dice there is 1 in three chance that there will be a square number if rolled, or a two in three chance it will not be rolled. rolling two dice at once is the same as rolling one dice twice so the probability the probabilty that she DOESN'T roll a square number is:

\(\frac{2}{3}\times\frac{2}{3}\)

which is, rounded off, 0.4445 or 44.45%

now the chances that she DOES roll a square number is 1 - 0.4445 which is 0.5556 or 55.56%

so the chances that she does roll a square number is 55.56% , which means that her rolling a square number is slightly more likely than not!

Scruffy23: I have a challenge for you !!. Work out the square root of 165 using "Newton's Method" to at least 5 decimal places. You can use a calculator but NOT the "sqrt key".

Let the  area  =  L * W    and the perimeter is  2(L + W )

So

Area Units  =  Twice Perimeter Units

 (L * W)  =   2 [ 2 ( L + W)]

L * W   =  4L  + 4W

L * W - 4L  =  4W

L (W - 4)  =  4W   ⇒     W  >  4

L  =  [ 4W ]  /  ( W  - 4)

Possible Values of  W  and  L

W          L            Ratio  of  [4W ] / (  W  - 4)  =    L

5          20                       20

6          12                       12

8           8                          8

12         6                          6

20         5                          5

Note that  the  last two dimensions are exactly the same as the first two...it just depends on the orientation of the rectangle

Note that   as  W   ⇒  infinity,   the ratio of  [4W] /  (W - 4)  ⇒ 4   but is always  > 4

So.....   no other integer values are possible  for  L

So.....the  possible dimensions are :

5, 20      6,12     and 8,8     

So....the sum of the possible areas is

5*20  +  6*12  +  8*8  =

100  +  72  +  64  =

100  +  136   = 

236

Geez, that was an extremely detailed answer. Thanks, Hectictar! I've learned an easier way to do it though.

First I convert radians to degrees.

\(\frac{\pi}{3}*(\frac{180^o}{\pi})\)

Multiply.

\(\frac{\pi*180^o}{3\pi}\)

The pi cancels.

\(\frac{180^o}{3}=60^o\)

So it is sin(60^o), and since I know the values of sin, cos, and tan of 30^o, 45^o, and 60^o, I know that \(sin(60^o)=\frac{\sqrt{3}}{2}\).

(a)

well since they can lay 7 miles of track every day, the formula i make for the amount of track there is left to lay is:

\(l=196-Ld\)

where l is the length of track remaining, L is the length of track they can lay in one day and d is the number of days.

so we replace L for 7 and d for 17 and we get:

\(l=196-7\times17\)

whichworks out to be 77 miles of track remaining to lay.

(b)

well since we want them to have 0 length of track remaining, and they can lay 7 miles of track a day, we can work out d. let's give the formula a bash to get:

\(d=\frac{196-l}{L}\)

we then replace l with 0 and L with 7 and get:

\(d=\frac{196}{7}\)

which works out to be 28. so it will take 28 days or 4 weeks for the railroad crew to lay 196 miles of track.

Simplify the following:

abs(-12 i + 5) + abs(-4 i + 3)

Absolute value gives the distance to the origin.

abs(5 - 12 i) = sqrt(Re(5 - 12 i)^2 + Im(5 - 12 i)^2) = sqrt(5^2 + (-12)^2):

sqrt(5^2 + (-12)^2) + abs(3 - 4 i)

sqrt(25 + (-12)^2) + abs(3 - 4 i)

sqrt(144 + 25) + abs(3 - 4 i)

sqrt(169) + abs(3 - 4 i)

Simplify radicals.

sqrt(169) = sqrt(13^2) = 13:

13 + abs(3 - 4 i)

Absolute value gives the distance to the origin.

abs(3 - 4 i) = sqrt(Re(3 - 4 i)^2 + Im(3 - 4 i)^2) = sqrt(3^2 + (-4)^2):

sqrt(3^2 + (-4)^2) + 13

sqrt(9 + (-4)^2) + 13 = 18

Let  m   be the missing term  and we have that

100x^2  - 36 =  4(5x + m) (5x - 3)       divide both sides by 4

25x^2  -  9   =   (5x + m ) (5x - 3)        expand the right side

25x^2  -  9  =   25x^2  + 5mx - 15x  -  3m   subtract  25x^2  from both sides

-9  =  5mx   - 15x   -  3m

-9  =  (5m - 15)x   -  3m

Since there is no x term  on the left side, we can equate this

0x  =  (5m - 15)x

So

5m - 15  = 0       add 15 to both sides

5m  = 15    divide both sides by  5

m  = 3

5y(2x + 7)  +  4(2x + 7)

Notice the common factor in both terms.

5y(2x + 7)  +  4(2x + 7)

Factor  (2x + 7)  out of both terms.

(2x + 7)(5y + 4)

the set of all points between point F and point G, including point F and point G

slope   =   [difference in  y  values] / [difference in  x  values]

2   =   [k - 3] / [5 - 1]

2   =   [k - 3] / 4

                               Multiply both sides of the equation by  4 .

8   =   k - 3

                               Add  3  to both sides of the equation.

11   =   k

To find  sin( pi/3 ) ,  we can start with an equilateral triangle with side length  1 .

Now draw a line that bisects an angle and the opposite side. The length of this line is sin(pi/3) .

By the Pythagorean theorem...

(1/2)²  +  ( sin(pi/3) )²   =   1²

                                                       Subtract  (1/2)²  from both sides of the equation.

( sin(pi/3) )²   =   1²  -  (1/2)²

                                                       Take the positive sqrt of both sides.

sin(pi/3)   =   √[1²  -  (1/2)²]

sin(pi/3)   =   √[1  -  1/4]

sin(pi/3)   =   √[3/4]

sin(pi/3)   =   √3 / 2      

Another approach....let x be the length of each shelf   and let 18 in  =

18/12 ft  = 3/2  ft =  1.5 ft

So we have

[ 3* 8   -  1.5  -  3x ]  =  .25  [ 3*8]     simplify

[ 24  -  1.5 - 3x ]  =  .25 * 24

22.5 - 3x  =   8         add 3 to each side, subtract 8 from both sides

22.5 - 8  = 3x

16.5  =  3x       divide both sides by 3

5.5  =  x   ⇒      5.5 ft  = the length of each shelf

x^3  + 8y^6

We have a sum of cubes here

a^3  + b^3   =   (a + b)  (a^2  - ab +  b^2 )

Let  a  =  ∛x^3  =  x         and   b  = ∛ [8y^6]  = ∛ 8  * ∛y^6  =     2y^2

So we have

(x + 2y^2 )  (  [x]^2  -   x*2y^2  +  [2y^2]^2  )     =

(x + 2y^2)  ( x^2  - 2xy^2  +  4y^4)    ⇒    the first answer

Set each factor to 0 and solve

2x -  4  = 0      add 4 to both sides

2x  = 4    divide both sides by 2

x  = 2

4x - 5  = 0    add 5 to both sides

4x  =5    divide both sides by 4

x  = 5/4

2 is the answer.

6xy -3x -8y  + 4       factor by grouping

3x (2y - 1)  - 4(2y - 1)  

(3x - 4) (2y - 1) 

We are looking for two things that sum to 8  and multiply to 16

1 + 7       1 * 7   =   7     no good

2 + 6        2* 6  =   12    no good

3 + 5       3 * 5  =  15    no good

4 + 4        4 * 4  =  16           this is what we need

So.....we have

(x + 4)  (x + 4)  =  (x + 4)²