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\(\sum_{1}^{14}\)  2^n-1  = $ 163.83   

  Day 14 will exceed 100 dollars       (day 13 =$81.91 )

"After how many days the total amount of money in her bank first exceed 100 dollars?"

                       funny wording.....could be AFTER day 13 on day 14 ??          

This is the equation of a circle...put it into standard form to find the radius....then area = pi r^2

x^2 - 8x        + y^2-9y        =   - 105      ' complete the square' for x and y

(x-4)^2          +  (y-4.5)^2  =  - 105  +16 + 20.25            ( are you SURE that was a  - 105 ??)

                                        I will assume -105 is supposed to be  - 10.5

                                          - 10.5 + 16 + 20.25 = r^2    r = 5.074 units

Then   pi r^2 =   25.75 pi units²

40 hours * 20.50 

   plus

10 hours * 20.50 * 1.5

   plus

3 hours * 20.50 *  2

All of that added  is her per week pay.....then multiply by 10 weeks to get her season pay total ........

40 x 20.50 =820 dollars per week - her regular pay

53 - 40 = 13 hours overtime per week

10 hours x 1.5 x 20.50 =307.50 @ 1.5 times her hourly rate.

13 - 10 = 3 hours @ double her hourly rate

3 x 2 x 20.50 =123 dollars per week @ double her hourly rate

820 + 307.50 + 123 =1250.50 dollars - she earned in 1 week.

1250.50 x 10 weeks =12505 dollars - she earned for that 10-week period.

0.424242424242......... can be expressed in lowest as m / n and NOT as mn

0.4242424242.....=14 / 33

m + n = 14 + 33 =47

Here is my attempt at your question:

p/y =2/3
2/5 * N =2/5N pink
3/5 * N =3/5N yellow

p/y = 3/4
3/7 * N =3/7N pink
4/7 * N =4/7N yellow

4/9 * M =4/9M packages A
5/9 * M =5/9M packages B

(4/9M * 2/5N) + (4/9M * 3/7N) =116 / 315NM
(5/(9M) * 3/(5N)) + (5/(9M) * 4/(7N)) =41 / 63NM

116 / (315NM) / 41 / (63NM) =7308 / 12915

Total Ratio of: Pink yarn balls / Yellow yarn balls
7308 / 12915 = 116 / 205

Using Pythagorean theorem:

1/  find BC     (AD = BC)

2/  find CF

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

3/  find all 3 angles of an isosceles triangle CFD    (∠CFD ≅ ∠FCD)

side AB = CD = (CF/2) / cos(FCD)

"The space between the circle and the inner square is what?"  <======= From the header of the question....implies AREA.

If you just want the length of the line, I did most of the work already:

    (circle diameter - square side)/2  =   (8 - sqrt32)/2      Nothing hard there.....anyone should be able to get that !

What question did you answer?!?

4/9 x  =  5/6 +2

4/9 x = 17/6

x = 17/6  * 9/4 = 153/24 =6.375

x = -y^2 -6y+12    this is a sideways parabola opening the left

    y coordinate of the vertex is   - b/2a    =     -  -6/(2(-1)) = - 3

    x coordiante will be found b plugging this y value into the equation   x = 21

Just divide each term by 4.

Yes, he is right!!!

big square side  a = 8          circle diameter   D = 8         small square diagonal   d = 8

small square side   s = sqrt(d² / 2) = sqrt(32)

The length of the arrow   z = (d - s) / 2    or      z = (8 - √32) / 2

Circle diameter = 8      radius = 4 cm    area = pi r^2 = 16pi   cm²

DIAGONAL of smaller square = 8 cm

   side length of smaller square    s^2 + s^2 = 8^2     s = sqrt 32

       area of square = s x s = 32 cm2

The FOUR areas inside the circle and out side the square =   16pi - 32    cm^2     

    if you want the area of ONE....divide by four

Sorry, there is a typo, here is the corrected version:

A, B, C, and D are standing in a circle. A has 800 mL of  100% juice in his glass. B, C, and D have 400 mL, 200 mL, and 100 mL of water in their glasses, respectively.

A pours half of his juice into B's glass and drinks half of what's left in his glass.

B then pours half of what's in her glass into C's glass and drinks half of what's left in her glass.

C then pours half of what's in his glass into D's glass and drinks half of what's left in his glass.

D then pours half of what's in her glass into A's glass and drinks half of what's left in her glass.

Lastly, all remaining beverages are poured into a big container.

The liquid in the big container then contains x% juice. What is x?

See if you can apply the method used HERE:

https://web2.0calc.com/questions/quadratic_76

Say the number of pencils is \(x\) and the number of pens is \(y.\)

We get, 

\(\begin{align*} \frac{y}{x} = \frac{3}{4} \\ \frac{y-5}{x} = \frac{2}{3} \\ 4y = 3x \\ 3y-15 = 2x \\ x = 60 \\ y = 45 \end{align*}\)

Therefore, the number of pencils is 60. (the equation is reasonably easy to solve with substiuation. 

Can you explain more?

The ratio of the width to length is sqrt(7)/2.

1. By power of a point, XY = 7*sqrt(3).

2. By power of a point, the minimum length of CQ is 6.

HCF =1...makes sense.....Thank you sooo very much again, Melody..Have a good night..