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Hint: Are you familiar with your IQ (intelligence quotient)?

GA

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Actually, the answer is 22 because polygons must have at least three sides. Nevertheless, your reply helped me figure it out! Thank you!! 

Lol! Such an obvious homework cheat!

i still get 1.8

this is confusing but thank you to anyone that helps!

Simplify as

4x^2 + 8x + c

To have a perfect square, the discriminant must =  0

So

8^2 - 4^4 *c = 0

64 - 16 c  =  0

16c = 64

c = 64/16 =4

The binomial is   (2x + 2)^2

Oh Holy The0neXWZ, whose answers are right and clear, what befuddled reasoning brings your AoPS question here?

I assure you your intellect is too wide and vast, to be troubled by such a little ask-

Yet still you ask, ever unsure, needing "help" on every turn,

Why don't you make it loud and clear, like all of your AI answers here-

What do you intend with our answers?

What will you learn from our laborious endeavours?

The answer should be obvious enough-

You copy your answer into the answer box, without giving so much a thought,

To the time and energy you just wasted

Even though they are sluts and simps-

CPhill, Bosco, Melody, ElectricPavlov, History, Plaintainmountain-

They have lives to live too,

Away from the problems of you,

And do them a favor, may I request,

Next time your integrity fails,

Do not come here to fret and wail,

Ask your good friend, whose answers are right and clear,

ChatGPT- whose name strikes fear.

GA

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EF  =    sqrt (2^2 - (sqrt 3)^2) =  sqrt ( 4 -3) =  sqrt (1) =  1

Area of triangle  FBE = (1/2) (AB) (EF) = (1/2)(2sqrt 3) ( 1) =  sqrt (3)

tan (angle  BFE)  =  sqrt (3) / 1  = sqrt (3)

arctan (sqrt (3)) = 60° = measure of angle BFE

So  angle AEB = 120°

Area of sector AEB =  pi * 2^2 / 3  = (4/3)pi

Area between sector AEB and triangle FBE =  (4/3)pi - sqrt (3)         (1)

Area  of small semi-circle  =   pi (sqrt 3)^2 / 2 =  (3/2) pi                     (2)

Shaded area =  (2) - (1)  =   (3/2) pi - [ (4/3) pi - sqrt (3) ]  =   pi/6 + sqrt (3)

To compare the two accounts, we'll calculate the final balances for both Account A (compounded monthly) and Account B (continuous interest) after 10 years.

Given:
Principal amount (initial investment), P = $5000
Time, t = 10 years

(a) Let's start with Account A:

Account A offers 2.3% annual interest compounded monthly. The formula to calculate the final balance with compound interest is:

\[A = P \times \left(1 + \frac{r}{n}\right)^{nt}\]

Where:
A = Final balance
P = Principal amount
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years

For Account A:
Principal, P = $5000
Annual interest rate, r = 2.3% = 0.023 (as a decimal)
Compounded monthly, n = 12 (times per year)
Time, t = 10 years

\[A_A = 5000 \times \left(1 + \frac{0.023}{12}\right)^{12 \times 10}\]

Now calculate \(A_A\).

(b) For Account B, which offers continuous interest, the formula for calculating the final balance is:

\[A_B = P \times \gamma^{rt}\]

Where:
A_B = Final balance for Account B
P = Principal amount
r = Annual interest rate (as a decimal)
t = Number of years
\(\gamma\)= Euler's constant (approximately 2.9234)

For Account B:
Principal, P = $5000
Annual interest rate, r = 2.3% = 0.023 (as a decimal)
Time, t = 10 years

\[A_B = 5000 \times \gamma^{0.023 \times 10}\]

Now calculate \(A_B\).

Compare the values of \(A_A\) and \(A_B\) to determine which account will yield the greater balance at the end of 10 years.

Once you've determined the greater balance, you can calculate the difference between the balances to find out how much more money Greg earns by choosing the more profitable account.

Let's use the given information to solve this problem. We know that the ratio of pencils to pens sold is 4:5. This means that for every 4 pencils sold, 5 pens are sold.

If the store sold 28 pencils, we can set up a proportion:

Pencils sold / Pens sold = Ratio of pencils to pens

Substituting the values:
28 pencils / Pens sold = 4 pencils / 5 pens

Now, solve for the number of pens sold:

Pens sold = (28 pencils * 4 pens) / 4 pencils
Pens sold = 33 pens

Therefore, the store sold 33 pens.

To determine whether an operation is defined or not for two matrices, we need to check whether the matrices have the same dimensions (same number of rows and columns) so that the operation can be performed. Let's analyze each option:

a.) L - N: Matrix L is a 2x2 matrix and matrix N is a 3x2 matrix. Since the number of columns is different, the operation is not defined.

b.) M - N: Matrix M is a 2x2 matrix and matrix N is a 3x2 matrix. Again, the number of columns is different, so the operation is not defined.

c.) Q + P: Matrix Q is a 1x1 matrix (scalar) and matrix P is a 2x1 matrix. The number of columns is different, so the operation is not defined.

d.) M + P: Matrix M is a 2x2 matrix and matrix P is a 2x1 matrix. The number of columns matches, so the operation is defined.

Based on the analysis:

Operation is defined (d.) M + P) 

Operation is not defined (a.) L - N), (b.) M - N), (c.) Q + P)

Partition of 10 cookies into 6 as follows:

1 = (4, 2, 0)
2 = (3, 3, 0)
3 = (2, 4, 0)
4 = (4, 1, 1)
5 = (3, 2, 1)
6 = (2, 3, 1)
7 = (1, 4, 1)
8 = (4, 0, 2)
9 = (3, 1, 2)
10 = (2, 2, 2)
11 = (1, 3, 2)
12 = (0, 4, 2)
Total number of ways== 12 ways to pick 6 cookies out of 10

There are a total of 4+4+2=10 cookies. We want to pick 6 of them, so the total number of ways to do this is (610​)=10!/6!4!​=210.

However, we need to divide this number by 3! because the order in which we pick the cookies does not matter. (For example, picking a chocolate chip cookie, then an oatmeal cookie, and then a peanut butter cookie is the same as picking a peanut butter cookie, then an oatmeal cookie, and then a chocolate chip cookie.) So the final answer is 210/3!=35​.

The denominator  simplifies  to   sqrt  [ 6 - 5x  ]

Since  we  can't  take  the  sqrt  of   a negative and  the  denominator  must not = 0  we have that

6 - 5x  >   0

6 >  5x

(6/5) > x     or

x <  (6/5)

Domain =  ( -inf, 6/5)

Second one

The blue area forms a square

Area of dodecagon  =  10 (1/2) (1/2 * diagonal of the  square)^2 * sin (30°)

1 =     5   *(diagonal of  the square/2 )^2  * 1 / 2

2/5 = (diagonal of the  square / 2)^2

sqrt (2/5)  = diagonal of the square / 2

Side of the  square  =    sqrt (2) * (diagonal of the  square / 2) =  sqrt (2) * sqrt (2/5) =  2 /sqrt 5

Area of blue square =  [ 2 /sqrt 5]^2  =   4/5

There are 52 cards in a deck of cards, of which 26 are black. So, the probability that two cards chosen at random are both black is 26/52​⋅25/51​ = 25/102​.

There are 51 pairs of adjacent cards in a circle of 52 cards. So, the expected number of pairs of adjacent cards that are both black is 51⋅25/102​ = 25/2.