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There are 30 outcomes. 10 + 8 + 12 = 30

Thanks, tertre   !!!

B

7                25

A          D    12      C

AC =  sqrt ( 25^2 - 7^2]  =  sqrt ( 625 - 49)   =  sqrt (576) = 24

And since D is the midpoint of AC, then DC =  (1/2)24 = 12  = AD

And tan BDA =  7/12

And angle  BDC is supplemental to  angle BDA

So.....BDC =  180° - BDA

So.....letting BDA = theta  and using  a trig identity  we have

tan ( 180° - theta)  =  tan (BDC) =  - tan (theta) = - tan(BDA) =  -7/12

35 - 26 = 9

21 - 18 = 3

42 - 35 = 7

25 - 23 = 2

Matrix looks like [ 9 , 7 ,...]

                           [ 3 , 2 ,...]

D 

D =   (4,1)

Reflecting this across the x axis gives D' = (4, -1)

Finding D" is a little tricky

The slope of the line through (4, -1)  and perpendicular to  y = x + 1  = -1

The equation of this line is

y = -1(x -4)  - 1

y = -x + 4  - 1

y = -x + 3

The x coordinate of the intersection of this line and y = x + 1  can be found thusly

-x + 3 = x + 1

2x = 2

x = 1

And y = 2

And.....(1, 2) is the midpoint of (4, -1) and D"

So the find the x coordinate of D" we have

[ 4 + x] / 2 = 1    

4 + x = 2

x = -2

And the y coordinate of D" is

[-1 + y ] /2 = 2

-1 + y = 4

y = 5

So

D"   =   ( -2, 5)

Here's the graph to show this : https://www.desmos.com/calculator/0nx7czfr1d

Wow, guest. That is a lot of work. I think there is a more "compact" solution. (see below)

Is there a diagram? (my bad, i didnt know that a diagram was not needed, aka my trig skills = not that great)

Add eq 1 and 3

-x + 3z = 8   (4)

Multiply eq 2  by 3 and add it to eq 3

-11x +16z=37

Multiply (4) by -11 and add to (5)

-17z= -58     so     z=3        Sub into (4) to find  x = 1 

Sub these into any of the equations to find y

x+3y +2z =1

1 +3y +2(3) = 1   thus y = -2

In some cases, (2) also works because an equilateral triangle is a special type of isosceles triangle.

.....using dimensional analysis   18hr * $ 9.45 / hr = $ 170 .10   (don't forget units    )

Answer to Q1 is 1

The greatest common divisor of 60  72  and 36     is   12 boxes

18 mg  / 20 mg  x 100% = 90% RDA Niacin

Boyle's Law   PV = k

so

P1 V1 = P2 V2

P1/ P2  V1 = V2

20.5/80.5  (400 ) = 20.5*400/80.5 = 101.86 ml

Solve the following system:
{x + 3 y + 2 z = 1 | (equation 1)
-3 x + y + 5 z = 10 | (equation 2)
-2 x - 3 y + z = 7 | (equation 3)

Swap equation 1 with equation 2:
{-(3 x) + y + 5 z = 10 | (equation 1)
x + 3 y + 2 z = 1 | (equation 2)
-(2 x) - 3 y + z = 7 | (equation 3)

Add 1/3 × (equation 1) to equation 2:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x+(10 y)/3 + (11 z)/3 = 13/3 | (equation 2)
-(2 x) - 3 y + z = 7 | (equation 3)

Multiply equation 2 by 3:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x+10 y + 11 z = 13 | (equation 2)
-(2 x) - 3 y + z = 7 | (equation 3)

Subtract 2/3 × (equation 1) from equation 3:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x+10 y + 11 z = 13 | (equation 2)
0 x - (11 y)/3 - (7 z)/3 = 1/3 | (equation 3)

Multiply equation 3 by 3:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x+10 y + 11 z = 13 | (equation 2)
0 x - 11 y - 7 z = 1 | (equation 3)

Swap equation 2 with equation 3:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x - 11 y - 7 z = 1 | (equation 2)
0 x+10 y + 11 z = 13 | (equation 3)

Add 10/11 × (equation 2) to equation 3:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x - 11 y - 7 z = 1 | (equation 2)
0 x+0 y+(51 z)/11 = 153/11 | (equation 3)

Multiply equation 3 by 11/51:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x - 11 y - 7 z = 1 | (equation 2)
0 x+0 y+z = 3 | (equation 3)

Add 7 × (equation 3) to equation 2:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x - 11 y+0 z = 22 | (equation 2)
0 x+0 y+z = 3 | (equation 3)

Divide equation 2 by -11:
{-(3 x) + y + 5 z = 10 | (equation 1)
0 x+y+0 z = -2 | (equation 2)
0 x+0 y+z = 3 | (equation 3)

Subtract equation 2 from equation 1:
{-(3 x) + 0 y+5 z = 12 | (equation 1)
0 x+y+0 z = -2 | (equation 2)
0 x+0 y+z = 3 | (equation 3)

Subtract 5 × (equation 3) from equation 1:
{-(3 x)+0 y+0 z = -3 | (equation 1)
0 x+y+0 z = -2 | (equation 2)
0 x+0 y+z = 3 | (equation 3)

Divide equation 1 by -3:
{x+0 y+0 z = 1 | (equation 1)
0 x+y+0 z = -2 | (equation 2)
0 x+0 y+z = 3 | (equation 3)

x = 1         y = -2         z = 3

No tertre!!. It is much more involved than that. Look under her first post here:

https://web2.0calc.com/questions/help-18_10#r1

If we are looking for "distinct" permutations, it is a very different story.

Yes, you're correct! If you look at the other choices, we have just numbers and letters, number followed by a letter. Thus, (B) or the second option. 

We can use dimensional analysis: Therefore, we have \(18h*\frac{9.45}{h}=18*9.45=\boxed{170.1}\) .

We just find for 10P4. Thus, the answer is \(\frac{10!}{6!}=10*9*8*7=\boxed{5040}.\)

Using the letters in the word INNOVATIVE, find the number of permutations that can be formed using 4 letters at a time. Show your work or explain how you got your answer.

Melody to the rescue!!. My computer says there are 1,398 permutations! But, for the life of me, I have NO IDEA how it gets that odd number! If you could figure it out, that would be just great. Thanks.

This much I learned: 

There are 246 "words" that begin with "I"

There are 246 "words" that begin with "N"

There are 246 "words" that begin with "V"

There are 165 "words" that begin with "O, A, T, E" for each.

So: [3 x 246] + [4 x 165] = 1,398 "words"

Prob of senior boy: 9/36 --> 1/4

Prpob of senior girl 7/35 --> 1/5

1/5 * 1/4 = 1/20 (D)