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Solution for #2:

\({}\)

Solve this using the derangement formula:

\(!n = \left [\dfrac{n!}{e} \right] \qquad | \qquad \text { where [ ] is the nearest integer, and (e) is Euler’s Number ~(2.71828...).} \\\)

(5/6)  * (1/3)   =   5  / 18    of the pizza   is what he ate

Amount remaining  =

5/6  - 5/18  =

15/18  - 5/18  =

10/18  =

5/9 

Perimeter  =  2 (W + L) 

42  = 2(W + L)

21  =  W + L

21 - W  =  L

Area =  L * W

104  =  (21 - W) * W

104  = 21W - W^2

W^2  - 21W  + 104   =  0       factor

(W  - 13) ( W - 8)  = 0

Setting  both factors to 0   and solving  for  W  gives us the dimensions

13  and  8

Let the unknown amount of the 70% solution  be  =  x

And we have that

8 * 4%    +   x * 70%    =   (8 + x) * 50%       ......so....

8 * .04   +  x * .7   =  (8 + x) .5     simplify

.32  + .7x  =  4 + .5x         subtract .32, .5x  from both sides

.2x  = 3.68         divide both sides by .2

x  = 18.4 KCl  of the 70% solution

The rope  will be the hypotenuse of a 45-45-90 right triangle

So....the height of the  treehouse  = H =    the  distance form where the rope touches the ground to the tree

So  ....using the Pythagorean Theorem  we have that

H^2 + H^2   = 80^2

2H^2  = 6400      divide both sides by 2

H^2  = 3200      take the positive square root

H  =sqrt (3200)  = sqrt (2 * 1600)   =  40 sqrt (2) ≈  56.6  ft = height of treehouse 

Let one base  be  =   B

Let the other be  = B + 6

Area of a trapezoid  = 

(1/2) height * ( sum of bases)    .....so we have....

40  = (1/2) (4) * ( B + B + 6)

40 = 2 (2B+6)     divide both sides  by 2

20  = 2B + 6       subtract  6 from both sides

14  = 2B      divide both sides by  2

7  = B  =   one base

And the other base =  B + 6 =  7 + 6    = 13

That guy is wrong and posting random stuff, real answer is 228pi beacuse using pythagorean theorem, we get sqrt(48+9) is the radius of circle. Since the surface area of a sphere is 4pi*r^2, we get 228pi as our desired answer. My aops is doctorpi btw

Let us make some equations: where the first side=\(x-3\), the second side=\(x\), and third side=\(2(x-3)\).

\(x-3+x+2(x-3)=71\)

\(x+x+2x-6-3=71\)

\(4x-9=71\)

\(4x=80\)

\(x=20\)

We need to find the longest side, which is the third side, or\(2(x-3)\).

\(2((20)-3)\)

\(\rightarrow34\leftarrow\)

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Hi there...

Problem...
----------------

The "Drive-Through"-menu...: 
------------------------------------------

Nr.1. = 1 burrito and 1 coffee for $5.60.
Nr.2. = 2 burritos and 3 coffees is $12.55.
-------------------------------------------------
Determine the cost of one burrito?

-------------------------------------------------

We know the following...

----------------------------
(a+b)=5.60

((2a)+(3b))=12.55

----------------------------

I hope that helped a bit?

Kind regards

BizzyX

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[(1 + sqrt(5))^23 - (1 - sqrt(5))^23] / [2^23*sqrt(5)] = 28,657 - the 23rd Fibonacci Number.

a=1;b=0;c=0;p=0; cycle:d=a*100+b*10+c;if(b==3 or a==3 and c==3 or a==3 and b==3 or b==3 and c==3 , goto loop, goto next); loop:printd," ",;p=p+1; next:c++;if(c<10, goto cycle, 0);c=0;b++;if(b<10, goto cycle, 0);b=0;c=0;a++;if(a<10, goto cycle,0);print"Total = ",p

OUTPUT = 99 such numbers.

Desmos graph for both equations.

I beleive that the other person over-counted.

Numbers of the form:   *3*  --->  90    [don't count numbers starting with a '0']

Numbers of the form:   3*3  ---> 9      [actually, there are 10, but '333' has already been counted]

Numbers of the form:   *33  and  33* have already been counted.

So, I think that there are only 99.

\(27 \div \frac{3}{4}=27 \cdot \frac{4}{3}=9\cdot4=\boxed{36}\) 

Its ok, there are many other ways to do that problem (eg. completing the square, factoring),  just remember the formula is \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) :D

You're welcome :D

We can break into cases:

333 - 1 number

33* - 10 numbers

3*3 - 10 numbers

*33 - 10 numbers

*3* - 100 numbers

Total of 131 sticky numbers.

42-35=7

at most 7 cookies.

https://web2.0calc.com/questions/please-help_67599

\(5/6*6/5=1\)

\(1/3*6/5=6/15\)

Mr. Maddox ate 6/15 slices of a whole pizza.

\(5/6=45/90 \)

\(6/15=36/90\)

\(45/90-36/90=8/90\)

\(8/90=4/45\)

4/45 of a pizza was remaining after Mr. Maddox ate some.

I don't know if I am right lol.

Should i label other angles a different variable (ig. angle p is y angle q is z)

https://www.eskesthai.com/2004/12/inverse-fourth-power-law.html

Multiply 1/3 by 2 to get 2/6 to make the denominator even and then do 5/6 - 2/6 it equals 3/6. When you multiply 1/3 by 2 do this 1/3 times 2/2.

This is considered an off-topic question.

The answer is Decibels (db).

"Sound is measured in units called decibels (dB). The higher the decibel level, the louder the noise. On the decibel scale, the level increase of 10 means that a sound is actually 10 times more intense, or powerful." credited to earq.com.

27 divided by 3/4

equal to 27 times 4/3

which is 108/3 or 36