All Answers 358468 Answers

\((x+y+z)^2 = x^2+y^2+z^2 + 2xy+2xz+2yz\\~\\ 0=\dfrac{(x+y+z)^2}{x^2+y^2+z^2}=\dfrac{2xy+2xz+2yz}{x^2+y^2+z^2}+1\\~\\ \dfrac{xy+xz+yz}{x^2+y^2+z^2}=-\dfrac 1 2\)

Hmmm....even if you cut the pictures into 1 inch by 1 inch squares, there's just not enough square inches on the board to fit 70 pictures worth.

number of square inches on the board  =  32 * 32  =  1024

number of square inches of 70 pictures  =  70 * 3 * 5  =  1050

Are you sure it says  70  and not  60  ?

If so, maybe you can be the one to report the error in this question. Yes it is weird and unlikely they'd have it wrong, but it's not impossible.

I don't know guys. Out of all the questions I've looked at made by this organization, their answers are always correct. Also, this was on a math contest taken by all middle schoolers in the U.S., so if the answer is wrong, then a student has to have already told them and they would have fixed it already. I also got 68, but these questions tend to have a creative way to solving them. Still, you guys have really great solutions and I agree with your reasoning why 68 is the maximum possible.

___ Increased by 130% is 69  

The answer will be the value of x in the following:

x + (130% times x)  =  69          increased by 130% means add another 130% to itself

x + 1.3x  =  69                           note that 130% = 1.3

2.3x  =  69 

x  =  69/2.3  

x  =  30

I can't see the drawing because it says "You need permission". Could you please publish it to the web or make it so that anyone with the link can see it? Thanks in advance.

Shouldn't there be two values for x, the smallest and the largest, to find the answer?

HAHAHA!!!!......yer right, EP......too much free time.....

I still say Logic Long Di c k sounds like a Di c k Tracy character.

When are you bringing in CPhill’s character?

Hey Rom.....   that is only a 30% increase......

The ball hits the ground when y  =  0....so.....

-4.9t^2 + 42t + 18.9  = 0        multiply through by  -10   to clear the decimals

49t^2 - 420t - 189  = 0         factor, if possible

(7t  - 63) ( 7t + 3)   = 0   set both factors to 0  and take the positive result

7t - 63  = 0                                                   7t + 3  = 0                        

7t  = 63       divide both sides by 7               7t  = - 3

7  = 9  sec                                                      t = -3/7 sec  (reject)

Here's a graph that verifies that the object hits the ground at 9 sec  after launch

https://www.desmos.com/calculator/bppxhga5of

Wow....looks like CHris is a bit bored!  HaHa  ! 

When the ball hits the ground, y= 0    so substitute that in to the equation and solve for t

0 = -4.9t^2+42t  +18.9      Use Quadratic Formula    a = -4.9  b = 42    c = 18.9

\(t = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

t=−0.428571   throw out

t=9 sec

(1)    f(g (x) )  means :   put g  into f

Because the math is messy...I'm going to use WolframAlpha to simplify this  ...so f(g(x))  =

4(x - 5) (x - 2)            4 (x- 5)(x - 2)               4x^2 - 28x + 40

____________  =    ______________   =  ______________   (1)

3(x^2 - 6x - 7)           3 (x - 7) ( x + 1)            3x^2 - 18x - 21

The vertical asymptotes  are the x values that make the denominator  = 0  these are   x  = 7   and  x = -1

So....their  sum  =  7 + -1   =  6

(2) As x approaches pos/neg  infinity.....y approaches  the ratio of the coefficients on the x^2 terms in (1).....that is

y  =  4/3 ......and this is the horizontal asymptote  in both directions

See the graph here :  https://www.desmos.com/calculator/xg6a4ctnqz

PLEASE CONTINUE THIS ON THE ORIGINAL THREAD.

(I HAVE OPENED IT)

Continued here

          .714285

7  [  50

       49

      ____

         10

           7

        _____

           30

           28

           ___

              20

              14

             ____

                 60

                 56

                ____

                   40

                   35

                 _____

                      50

So   .714285... .≈   71.4285%

x^2 + y^2  = 14x + 48y         rearrange as

x^2 - 14x  + y^2 - 48y  = 0      complete the square on x and y

x^2 -14x + 49 + y^2 - 48y + 576  =  49 + 576        factor  and simplify

(x -7)^2  + ( y - 24)^2  =  625

This is a circle centered at  ( 7 , 24)   with  a radius of √625  =  25

The minimum value  for x  will be :     x coordinate of the center  minus the radius  =

7 - 25   =    -18

\(x \cdot \dfrac{130}{100} = 69\\ x = 69 \dfrac{100}{130} = \dfrac{690}{13} \approx 53.077\)

Since we have a vertical asymptote  at  x  = -2.....then  (x + 2)  must be a factor of q(x)

Since we have a horizontal asymptote at y = 2, then 2 must be a factor of p(x) since p(x) and q(x) have the same degree

Since we have a hole at x  = 5....then (x - 5)  must be a factor of p(x) and q(x)

Since the point (1,0)  is on the graph, then ( x - 1) must also be a factor of p(x)

So.....the function is

2(x - 5) (x - 1)

___________

(x - 5) ( x + 2)

And

p(3)          2(3 - 5)(3 - 1)           2 (-2)(2)          -8          4

____  =   ____________ =     ________  =   ___  =  ___

q (3)           (3 - 5) (3 + 2)         (-2) (5)           -10         5

So  (3, 4/5)  is on the graph

Here's a pic : https://www.desmos.com/calculator/gydybu0wek

Let x be the number

x + 1.3x = 69

2.3 x = 69

x = 69/2.3 = 30

x^2-7x <0

x(x-7) <0

so x<0    or x <7  but if x<0 the answer would be >0

so the only integers would be   1 2 3 4 5 6