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Thank you for your answer. How did you get from x^2 = 2i,-2i to the final answers?

Well A and B are our candidates

but the thing is that a will be a positive decimal whereas b will be >1

so b

How long is the sequence?   Only 3 letters      ?   is  ZERO A's considered a possibility?

AAx

AxA

xAA      where x can be   B or C        3 x 2 = 6 sequences

This means that y is a  negative fraction

A  when you square a negative fraction you get a smaller value positve fraction that is <  +1

B  when you subtract a negative fraction from 1   you get a number >1

C  when you add a negative fraction to 1   you get a positve fraction < 1

D   when you multiply 2 times a negative fraction ....you get a more negative number

TWO triangle hypotenuses      find via pythag theorem

    h^2 = 4^2 + 10^2        solve for h    rememeber: there are 2

semicircle perimeter =   1/2  pi d   = 1/2 pi * 8         add them together

\(x = {-4 \pm \sqrt{4^2-4(3)(12)} \over 2(3)}\\ x = {-4 \pm \sqrt{-128} \over 6}\\ x = {-4 \pm 3\sqrt{-2} \over 6}\\ ({-4 + 3\sqrt{-2} \over 6})^2+({-4 - 3\sqrt{-2} \over 6})^2\\ {(-4 + 3\sqrt{-2})^2 \over 6^2}+{(-4 - 3\sqrt{-2})^2 \over 6^2}\\ {(-4 + 3\sqrt{-2})^2 \over 36}+{(-4 - 3\sqrt{-2})^2 \over 36}\\ {(-4 + 3\sqrt{-2})^2+(-4 - 3\sqrt{-2})^2 \over 36}\\\)

Well this is kind of bashy, lOl.

\(r + s = -\frac{4}{3} \text{ due to vieta}\\ (r + s)^2 = \frac{16}{9} \text{ because we square -4}\\ r^2+s^2+2rs=\frac{16}9\\ rs=4 \text{ due to vieta}\\ r^2 + s^2 = \frac{16}{9} - 2(4) = \frac{16}{9}-8=-6\frac{2}{9}\)

Someone check this

Ahh, I see how you're doing it. Yes, guest, you can also do it in cents or just multiply by a 100 if you want cleaner numbers to work with. Nice catch!

Set:

\(n=\text{number of nickles}\\ q=\text{number of quarters}\)

Now, we can make a system of equations. We have:

\(\begin{align*} q+16&=n\\ 5.90&=0.05n+0.25q \end{align*}\)

I trust you can solve this yourself. If not, here it is even more simplified. Substituting \(n\) in, we have:

\(5.90=0.05(q+16)+0.25q\\ 5.90=0.05q+0.8+0.25q\\ 5.90-0.8=0.3q\)

Now finish on your own! :)

q = quarters                  value = 25q

n = nickles =  (q+ 16)    value  =   5 (q+16)      values summed = 590

25q      +    5(q+16)       = 590      solve for   q

multiply  r^5   x 16   to get to 24

      24 =  16 r^5        r = \(\sqrt[5]{24/16}\)

\(\sqrt[5]{24/16}\) )²⁰  * 16 = 81

Hmm.  I'm afraid the answers here are wrong!  The original clearly says "diagram not to scale".  The diagram is drawn as it is to be misleading! If you don't believe me try to draw it using the specified lengths and angles.

Here is the correct answer:

Well, try to find the difference. We have 5 terms from 16 to 24, and the difference is 8. so, each step must be increasing by 8/5. This gives the order:

\(23rd: 16\\ 24th: 16\frac{8}{5}\\ 25th: 16\frac{8\cdot2}{5}\\ 26th: 16\frac{8\cdot3}{5}\\\)

We have a pattern, and that is that the answer for a given sequence is for any term \(n\), we have \(16+\frac{8*(n-23)}{5}\). Try numbers out! And most importantly, plug 43 into that expression. What do you get?

Well, think about it! A slope of seven means that the \(\frac{\text{change in y}}{\text{change in x}}=7\). So, using our formula, we can make an equation from your points, giving\(\frac{2a-(-2)}{6-a}=\frac{2a+2}{6-a}=7 \). See if you can solve the equation for yourself now!

* deleted *    Question misread ...

I believe you made a typo, jimkey.  Look at what you said.  

The total area is given as 6. 

You said that a smaller area inside the total area is double that much. 

_.

Calculate the circumference of the complete circle. Take 3/4 of that and add to it the lengths of OP and OQ (which are radii of the circle).

Oh I forgot sorry,  Here is the diagram 

y = -1/2 x + 3/2       and     y = -5x-1       equate the two equations

-1/2x + 3/2   = -5x -1       Now solve for 'x'

                      use this valueof 'x' in one of the equations to find the corresponding 'y'

Central angle covers an arc of 80 ^o     leaving c covering  360-80 = 280 degrees

     angle c = 1/2  * 280 = .........   

This was a fun problem for people to try out, as I thought it was interesting. I think it is interesting, because many people get the right answer, but the method of solving is incorrect for many people. I was saying that the answer was correct! It is off-topic, so I didn't need it for anything. 

Amount she owes   compounded quarterly    6000 ( 1 + .12/4)^(4*4)  =..........

Amount she owes   compounded annually     6000(1+.12)⁴ = .........    

 I'll let you figure out the DIFFERENCE......

Thank you so much!!!

a   .05 x   =  1                    x is the years to double

b    (1.05)^x =2                    x is the years to double

c     (1+ .05/2)^2x = 2          x is the years to double

d     (1 + .05/12)^12x = 2     x is the years to double

that means that we can seperate the hexagon into 18 equal traignles with each area 1/3 and muliply it 12 since there are 12 of the triangles that fit exactly in the star so the area of the star is 12*1/3=4