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I think the different perimiters part is the key to solving this problem efficiently...

The maximum perimiter that any rectangle can have is 40, and you can only draw one of those.

The next biggest perimiter would be 39, but you can't draw a rectangle with integer side lengths with that perimiter.

The next one would be 38, you can draw lots of rectangles with that peremiter, but only one will count because they have to have different perimiters.

37 is out because its odd

36 is in because its even

all the way down to a peremiter of 4, which is the smallest rectangle that meets the restrictions

There are 40/2 - 1 = 19 possible rectangles.

Yeah, post the image properly, please. Just posting the asymptote in the question box won't show up.

Redacted: Check the side lengths.

The Subject of the letter was to tell the Clergymen his reason for why he had come to Birmingham. In the letter, (comma) he stated that he was here to achieve equal rights for the people whom who were oppressed. He also stated that he was going to use nonviolent means such as protesting negotiation to achieve his goal of equality.

Occasion:
Martin Luther King Jr. Wrote this letter inside Birmingham jail because he was accused of his improper acts of civil disobedience. Dr. King was doing these protesting techniques in order to Sentence is too wordy, consider changing "in order to" into "to" express his disappointment at the people and laws who were suppressing the black community. He wanted to fix the unjust actions that were implemented into "in"society. 

Purpose:
The purpose of writing the letter was to respond to the publishing in the local newspaper thomthat was posted by eight clergymen in Alabama. The Newspaper that the Clergymen posted was asking Black’s to not change the current way of life, to stay oppressed and submissive to the white man. Dr. King’s response to the statement was that Without action no result would occur and so he wants them to take a stance without violence as he believes that only escalates the situation. It also explains his reasons for starting the civil disobedience. He wants to change the way that people are treated all over the country and explains the horrible way that African Americans are being treated. 

Speaker
Martin Luther King Jr. is the speaker in the letter. He was a civil rights activist for many years and wanted to see that people should all be treated equalequally. 
 
Tone
The tone that is conveyed in the letter shift and change in order to match the words he is conveying at the time. He keeps a courageous and strong will throughout the speech and includes dedicationa dedication in areas such as his willingness to suffer for equality among people. He never changes the tone of the letter to anything of violence or hate. He always keeps a passive stance, a very peaceful one in order to achieve his ultimate goal of equality among men in the country. The tone of the letter has also been changed to that of previous speeches and addresses. In which this he writes this to convey it to his audience which was primarily Reverends and Pastors. Due to this limited religious practices, (comma) he includes many religious ideals in the letter to address these men personally. 

Here is a hint on how you may check your answer:

If the perimeter of your square is  40, then each side is 10.

This means you can draw a 10 by 10 square and imagine 100 small equal squares inside.    Now try to find all the possible unique rectangle shapes.

(check your answer again there should definitely be more than twelve.

Thanks, tertre and heureka.....those "formulas"  provide a nice shorthand way to solve this type of problem  !!!!

What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 5:44?

\(\text{Let the angle formed by the minute hand and hour hand in degrees $\mathbf{ \Delta \alpha }$ } \\ \text{Let the time in hours $\mathbf{ t }$ }\)

The formula between the two values \(\Delta \alpha\) and \(t\) is:
\(\boxed{~ \Delta \alpha = 330 \cdot t } \)

\(\begin{array}{|rcll|} \hline \Delta \alpha &=& 330 \cdot t \quad | \quad t= 5:44 = 5+\dfrac{44}{60}= 5.7\overline{3}\ h \\ &=& 330 \cdot 5.7\overline{3} \\ &=& 1892 \\ &=& 1892 -5\cdot 360 \\ &=& 1892 - 1800 \\ \mathbf{\Delta \alpha} &=& \mathbf{ 92^\circ } \\ \hline \end{array}\)

Let a, b, c be real numbers such that \(a + b + c = 1\).

Find the minimum value of \(2a^2 + 3b^2 + 6c^2\).

Schwarz Inequality:

\(\begin{array}{|rcll|} \hline \mathbf{(x_1^2+x_2^2+\ldots + x_n^2)(y_1^2+y_2^2+\ldots + y_n^2) } &\geq& \mathbf{ \left(x_1y_1+x_2y_2+\ldots x_ny_n \right)^2 } \\\\ \left( \dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6} \right)(2a^2+3b^2+6c^2 ) &\geq& \left( \dfrac{1}{\sqrt{2}} \sqrt{2}a+ \dfrac{1}{\sqrt{3}} \sqrt{3}b+ \dfrac{1}{\sqrt{6}} \sqrt{6}c \right)^2 \\\\ \left( \dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6} \right)(2a^2+3b^2+6c^2 ) &\geq& \left( a+b+c \right)^2 \\\\ \left( \underbrace{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}}_{=1} \right)(2a^2+3b^2+6c^2 ) &\geq& \left(\underbrace{ a+b+c}_{=1} \right)^2 \\ 2a^2+3b^2+6c^2 &\geq& 1^2 \\ \mathbf{2a^2+3b^2+6c^2} &\geq& \mathbf{1} \\ \hline \end{array}\)

The minimum value of \(2a^2 + 3b^2 + 6c^2 \)is \(\mathbf{1}\)

Yep it is!

Thx!

Post the pic please, i think you posted the code of the picture not the real picture

You get points by posting questions or getting liked for your answers

i dont know exactly how it is done but yeah

thanks for the help!

Since Chris has answered after me I assume I made a silly mistake somewhere.

Thanks for doing as I requested.

Try to use Cauchy-Schwarz Inequality because of the sum of squares. 

You can also use this formula. \(|30h-5.5m|\), where \(h\) denotes the number of hours and \(m\) is the number of minutes. Thus, the answer is \(|150-242|=|-92|=92\) degrees.

You could take the picture by ctrl-shift-3 and upload it using the image function. Also I tried creating an account but it said server error. What's wrong?

It should be a comma rather than a semicolon. 

.

We actually  have the coordinates ( 1,3) (2,1) (3.5) (4,4)  (5,2)

The inverse reverses these  so we would have ( 3,1) (1,2) ( 5,3) (4,4) (2,5)

So  f^-1(1)  =   2

And   f^-1 ( f^-1(1))  =  f^-1(2) = 5

Finally

f^-1 ( f^-1 (f^-1 (1) ))   =    f^-1(5)  = 3

I like doing this in the following manner....construct a square with a side  = 4  [ any side length would work ]

Let A  = (0,0)  B  = (0,4)  C = (4,4)  and D  = (4,0)  E  = (2,4)  F  = (4,2)  M = (0, 2)

Here's a pic : 

Triangles BCF  and DAM are congruent right triangles

And the slope of BF  =  [ 4-2] / [ 0 - 4  ]=  2/-4  = -1/2

And the slope of EA =   [ 4-0]  / [2-0]  = 4/2  = 2

So....these segements have reciprocal slopes  so they are perpendicular

This means that triangles   BCF  and BGE  are similar by AA congruency

BC  = 4     CF  = 2     BE  = 2    BF  =  √ [ BF^2 + CF^2]  = √ [4^2 + 2^2] = √20  = 2√5

So.....by similar  triangles......  BG / BE   =  BC/ BF   →  BG/ 2  = 4/ [ 2√5 ]   →  BG  = 4/√5 

And GE  =   √[BE^2  - BG^2]  = √[2^2 - (16/5]  = √  4 - (16/5]  = √ [ 20 - 16] / √5  =  2/√5   

And the  area of triangle  BCF  =  (1/2) BC * CF  = 4

And since BG / BC  =  (  4 / [ √5] ) / 4   =  1/ √5.....the area  of triangle BGE  = 4 (1/√5)^2  =  4/5

Likewise  triangles MAD  and  MNA   are similar  and we can show that triangles BGE  and ANM are congruent

So....BG/MN  = BG / GE   =  [4/√5 ] / [2 /√5]    =   2

And FG  =  BF - BG   =     2√5  - 4/√5   =  [10 √5 - 4√5] / 5  =  6√5/5  =  6/√5

So....FG/ MN  =  6/√5 / [ 2/√5] =  6/2  =  3

And DN  = DM - MN  =   BF - GE =  2√5  - 2/√5  =  [ 10√5 - 2√5] / 5 = 8√5/5  = 8/√5

So DN / MN  = [8/√5]/ [2/√5]  = 8 /2  =  4 

Area  of BMDF  =  Area of the square  less areas  of BCF  and DAM  =  4^2  - 2 [4] = 16 - 8  = 8

Area of  GBMN  =  area of BEA - areas of BGE  and ANM  =  (1/2)BE * BA  - 2(4/5)  = 

(1/2)(2)(4) - 8/5   =

4- 8/5   =

[20 - 8] / 5  =

12/5

So the area of  GFDN  = area of BMDF  - area of GBMN  =   8 - 12/5 =  [40 - 12]/5 = 28/5

So  GFDN / GBMN  =  [ 28/5] / [ 12/5] =  28/12   =  7 / 3

Here I will give you hint:        EDIT: If you want you can go down to CPhill's answer for the answer

You can calculate the area of MAD, BEA, CFB.

You can calculate the area of BMDF by taking the area of the entire square and subtracting MAD and CFB from it.

You can do some coordinate geometry to find info about BGNM

Yes, given the info above. Assuming if the paper is graded SOLELY on the amount of words, then your answer is correct.

dont be scared of those fractions, just think of it as 1, 2, 3 ... 8, 9, 10.

If a set  has n elements, then it possesses  2^n   subsets   [including the empty set ]

To see this....consider that every element is either  in or out of a subset....so....each element has two " states of being"

So....for  n elements, we have 2*2*2*2.....  = 2^n  subsets

So.....we have 10  members of set A

So....it possesses  2^10   =   1024  subsets