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I was looking for the angle whose cosine value is for 2 units larger then the value of sine.

And these numbers popped up:    0.8 and 0.6        sin(36.87°) = 0.6     cos(36.87°) = 0.8

I multiplied these numbers by 10,  and got the sides of a rectangle: a = 6   and  b = 8

Diagonal   d = r                      r² = a²+ b²           r = 10      

We can drop perpendiculars from X to PQ and from Y to SR.  Then by applying the Pythagorean Theorem, we get an answer of XY^2 = 6^2 + 22^2 = 520.

You can do it by casework on the number of ways of distributing the balls into the boxes.  This gives you 1 + 3 + 2 + 4 + 1 = 11 ways.

Let the number of shirts sold =S

Let the number of subscriptions sold =N

N = S..............................(1)

5S - 40 =4N...................(2)       Sub (1) into (2)

N = 40 subscriptions sold

S = 40 shirts sold.

Check:

5 * 40 - 40 =$160 profit on selling 40 shirts

4 * 40         =$160 profit ftom selling subscriptions.

(XY)^2 = 4*(5^2 + 12^2)^2 = 676.

Not sure if this will work, but I found the GCF of 380 and 200 and divided 380 and 200 by the GCF.

The GCF is 20,

so its really just 19 cards and 10 persons.

Then I found the number of cards facing upward, and multiplied that by 20.]

Probably a really bogus solution.

See the full answer here: https://mindyourdecisions.com/blog/2018/11/01/challenge-for-13-year-olds-how-tall-is-the-bottle/

% means parts of 100, so:  \(41\text{%} \rightarrow \frac{41}{100}\)

"Given that m - 2 is a positive integer that divides 3m^2 - 2m + 11, find the sum of all such values of m."

\(\frac{3m^2-2m+11}{m-2}=3m+4+\frac{19}{m-2}\)

For the last term to be an integer we must have m - 2 = 1, so m = 3;  or  m - 2 = 19, so m = 21

Hence sum = 3 + 21 = 24

Ok thanks for your explanation.

I still do not even understand the question, 

But it doesn't matter.

Just add 5.4 each time:

–11.0 + 5.4  =  –5.6    

                        –5.6 + 5.4  =  –0.2         

                                              –0.2 + 5.4  =  5.2              

                                                                    5.2 + 5.4  =  10.6    

                                                                                        10.6 + 5.4  =  16.0

I'm not the same Guest as in the first answer that got it wrong.                 

_.

I think the sum is infinite:

***delete***

Find the area of a cyclic quadrilateral with sides 2, 2, 3, 1.

Formula: \(\boxed{ A = \sqrt{(s-a)(s-b)(s-c)(s-d)}\\ \text{where }s \text{ is the semiperimeter }\\s=\dfrac{a+b+c+d}{2} } \)

\(\begin{array}{|rcll|} \hline s &=&\dfrac{2+2+3+1}{2} \\ \mathbf{s} &=& \mathbf{4} \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline A &=& \sqrt{(4-2)(4-2)(4-3)(4-1)} \\ A &=& \sqrt{2*2*1*3} \\ \mathbf{A} &=& \mathbf{2\sqrt{3}} \\ \hline \end{array}\)

Point E is the midpoint of the side BC of triangle ABC and F is the midpoint of AE. The line through BF intersects AC at D. Find the area, in cm squared, of triangle AFD if the area of the triangle is 48 cm²

Your concept is good, but the answer is wrong!

You did not subtract the area of a triangle  CDG.

The area of AFD = 4 cm²   

Red area             A_r = 12*16 = 192 u²

Green area         A_g = (25 * 192)/6 = 800 u²

The area of entire rectangle  A_e = A_r + A_g = 992 u²  

Why?

The power of the  polynomial  in the numerator is less than that in the denominator.....so.....the limit  of this series will be  0

If x^2 + 1/x^2 = 171, what is the positive value of x - 1/x?

Hello Guest!

\(x^2+\frac{1}{x^2}=171\\ \frac{x^4+1}{x^2}=171\\x^4+1=171x^2\)

\(\color{blue}x^2=a\\ a^2+1=171a\\ a^2-171a+1=0\)

\(a=\frac{171}{2}\pm\sqrt{(\frac{171}{2})^2-1}\)

\(a_1=170.99415\\ x_{1,1}=13,0764732\\ x_{1,2}=-13,0764732\\ a_2=0.0058481532\\ x_{2,1}=0.0764732\\ x_{2,2}=-0,0764732\)

\(x_{1,1}-\frac{1}{x_{1,1}}=13,0764732-\frac{1}{13,0764732}\color{blue} =13\\ x_{1,2}-\frac{1}{x_{1,2}}=-13,0764732-\frac{1}{-13,0764732}=-13\\ x_{2,1}-\frac{1}{x_{2,1}}=0.0764732-\frac{1}{0.0764732}=-13\\ x_{2,2}-\frac{1}{x_{2,2}}=-0.0764732-\frac{1}{-0.0764732}\color{blue}=13\\ \)

The positive value of x - 1/x is 13.

  !
 

The RED area is 16 x 12 =192  u^2

  this represents   6 out of (25+6)    or  6 out of 31

6/31 x  = 192

x = 192 * 31/6 = 992 units^2 = total area