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We know an angle, the opposite side to the angle and we are looking for the hypotenuse.....the sine function relates these.....so we have

sin (65)  = 15 / d   rearrange as  

d  = 15 / sin (65)  ≈  16.6 ft

-32°  =  328°

Adding  360°  to this  gives us 688°

And  -32°  - 360°  =   -392°

arcsin (-1/2)   =  -pi/ 6

Conversion from rads to degrees  uses this  ⇒  180° / pi

3pi/ 4  x  180° / pi  =

3/4 x  180°   =    135°

I am willing to share another method for the eighth question

#8)

The method I default to uses the following conversion. I will create another table:
 

There is only one unknown here! We only need to find the number of grams that equals the number of moles of Cl₂. There are 35.45 grams of Cl per mole, according to the trusty periodic table. Cl₂ has double the number of molecules as Cl, so  \(1\text{molCl}_2=35.45g*2=70.9g\). In case you are unaware, "STP" is shorthand for standard temperature and pressure.

This method is somewhat different than the guest's because it does not need to consider the amount of Cl₂ reacted. The general conversion strategy for this particular procedure goes as follows. The table displays the general layout, and all unknowns are marked with question marks:

\(\text{grams}\rightarrow\text{moles}\rightarrow\text{moles}\rightarrow\text{grams}\)

Let's travel through this table column by column. We start with the given information, 16.2gH₂, and the eventual goal is to perform a series of conversions. The first column of the table is already finished. 

Yes...without  "b,"  all vertcal parabolas would have a vertex on the y axis....but, of course, the vertex can definitely be away from the y axis

Is b important then? My problem is with b.

a  =   the width of the parabola

If  l a l < 1...the graph is wider than the graph y = x^2, if  l a l > 1....the graph is narrower than the graph y =  x^2  and if a < 0....we have an "upside" down parabola

c  =  the y intercept

b...when this isn't 0,  it shifts the vertex of the graph off of the y axis....either left or right

Look at these graphs to get a feel for this :

https://www.desmos.com/calculator/shyvwfty0k

For the second one....the center of the circle forms the remaining vertex of the rectangle......and the circle will have a radius of 9

The intersection of the rectangular region  and the inside of this circle will just be the area of 1/4 of this circle....so it will be :

(pi/4) (9)^2   =  [81/4] pi   units^2

Using the form  

y  = a (x - h)^2  + k      where the vertex  is (h, k)   = (2, -4)      

We know that, since (4,12) is on the graph

12 =  a ( 4 - 2)^2  - 4

16 =  4a    ⇒  a  = 4

So we have

y  = 4(x - 2)^2  - 4

Expanding this we have

y  = 4x^2 - 16x + 12

To find the zeroes

0  = 4x^2 - 16x  + 12     divide through by 4

0  = x^2  - 4x  + 3         factor

0  = (x - 3) ( x - 1)

Setting each factor to 0  and soving for x we get that the roots are  x  = 3  and x = 1

So...  m  - n  =  3  - 1   =  2

A committee is to be made up from a group of 10 candidates. The committee must consist of a chairman and some (not necessarily nonzero) number of members-at-large. How many ways can the committee be chosen?

Chairman+0        10                                       =    10 ways

Chairman+1        10*9                                    =   90 ways

Chairman+2        10*9C2       = 10*36            =   360 ways         

Chairman+3         10*9C3      =  10*84           =   840 ways        

Chairman+4         10*9C4      =    10*126       =   1260 ways        

Chairman+5         10*9C5      =    10*126       =   1260 ways  

Chairman+6          10*9C6      =  10*84           =   840 ways

Chairman+7         10*9C7       = 10*36            =   360 ways

Chairman+8        10*9C8        = 10*9            =   90 ways       

Chairman+9        10                                      =    10 ways         

Now you can add up all the ways  :)       

x - y^2 + 6y  = 8      rewrite as

x  =  y^2 - 6y  + 8     

This one  is a little different  from the "normal" vertical parabola

It turns sideways and opens to the right

The y coordinate of the vertex is given by 

- (-6) /2(1)  =  3

So  the x coordinate of the vertex  is

x  =  (3)^2 - 6(3) + 8

x  = 9 - 18 + 8

x  = -1

So......the vertex  is    (-1, 3)

Here's a graph  ; https://www.desmos.com/calculator/wicxhszt9d

"Domain" refers to the x values covered by the graph....

The least x value  is at x  = -2 and from here, all x values > -2  are covered

So....the domain is  x ≥ -2

The solution to  f(x)  = g(x) is the point where the graphs  intersect.....this is at  ( -2, - 1 )

The first one is easy....the slope is negative  and the y intercept   is  3

In the form  y  =  mx  + b, m is the slope  and b is the y intercept

So...  the equation  y  =  -2x + 3   is the only one with a negative slope and a y intercept of 3

For the second...if -4, 1 and 2   are  solutions   to   0  = (x + 4)(x - 1)(x - 2)......so the graph intercepts the x axis  at   x = -4, x = -1  and x  = 2

"D" looks to be correct

Thank you soooo much!! 

I believe I have settled this one. The issue is interpretation-based. 

Cphill's interpretation: \(4\sqrt{15}-\textcolor{red}{\frac{10\sqrt{3}}{5}}+3\sqrt{60}\)

ManuelBautista2019: \(4\sqrt{15}-\textcolor{red}{10\sqrt{\frac{3}{5}}}+3\sqrt{60}\)

According to my individual calculation, both of you got the correct answer, assuming your interpretation.

The y intercept is (0, 5)....so  "c"  = 5

The vertex  is  (2,1)

The coordinate of the vertex  is

-b/[ 2a ]  =  2

-b = 4a

b = - 4a

And (1,2)  is on the graph....so we have that

2  =  a(1)^2  - 4a(1)  + 5

2  = a  - 4a  + 5

-3  = -3a

1  =  a

So   b = -4(1)  =  -4

a + b + c   =   1 - 4  + 5    =  2

f(x)  = x^2  - 5

When the domain is 2, the range is -1

When the domain is  4, the range is 11

When the domain is 7, the range is 44

"D"  is correct

x^2 + 16

The sum of two squares can never be factored in real number terms

V  = 2*pi*r^2*h     divide both sides by  2*pi* r^2

V / [ 2 * pi *r^2 ]  =  h =  "D"

Thank you CPhill and guest!

¸We have a total of 12 marbles....let x be the number of green ones we need to add

[ Green + Added Green ] / [ Total + Added Green ]  = .75  = 3/4

[ 5 + x ] / [ 12 + x ]   =   3/4    cross-multiply

4 [ 5 + x ] =  3[ 12 + x ]    simplify

20 + 4x  =   36 + 3x      subtract  3x, 20 from both sides

x  = 16 ⇒    16 green must be added

(-1,3)

See a tentative answer under your first question.

7)  Ok, I will take at stab at it, but you will have to have it checked by some expert!.

Since the molar mass of H_2 =2.016 g/mole, then 16.2 grams =16.2/2.016 =8.04 moles of H_2.

Since the reaction will require the same number of CL_2 moles, or 8.04 moles, it therefore will rquire:

8.04 x 70.9 =~570 grams of CL_2, because 1 mole of CL_2 =70.9 g/mole.

Therefore: 16.2 + 570 =586.2 grams of 2HCl

Thanks!!