All Answers 356462 Answers

is this from aops because i just saw this problem in one of the aops videos they provide in homework

We can split the triangle into 2 portions as shown below.

First let's find the coordinates of Point A and Point B, which are both on the same line of slope -3 going through point 4, 4.

\(y = -3x + b \\ 4 = -12 + b \\ b = 16 \\ y = -3x+16\)

Because Point B is the y-intercept of this line, Point B is at (0, 16). We can plug in y = 0 to find Point A.

\(y = -3x+16 \\ 0 = -3x + 16 \\ -3x = -16 \\ x = \frac{16}{3}\)

So Point A is at (16/3, 0). First we can find the area of the red.

\(16 \cdot \frac{16}{3} \over 2 \\ \frac{216}{3} \cdot \frac{1}{2} \\ \frac{216}{6} = 36 \)

The red section has an area of 36. Moving onto blue, 8 - 5 1/3 = 2 2/3 = 8/3. Height is equal to 4.

\(4 \cdot \frac{8}{3} \over 2 \\ \frac{32}{3} \cdot \frac{1}{2} \\ \frac{32}{6} = 5 \frac{1}{3} \)

36 + 5 1/3 = 41 1/3

thanks you <3

P is going to meet A and B at their midpoint, which is (27/2, 3/2). 2x - 4y = 27 - 3*2 = 21. 

A - 470 Y, 480 O
B - 350 Y, 200 O

So let's say after transferring the M&M's, box B has b yellow ones. That means box B also has 4b orange ones (because 0.8 = 4 times of 0.2). Meanwhile, box A will have a yellow M&M's and 2a/3 orange ones.  

We have 820 yellow ones total and 680 orange ones total. 

a + b = 820 
2/3*a + 4b = 680 

We can multiply the first equation by 4 to get 4a + 4b = 3280. Now subtract the second equation from this, and we find that 10/3*a = 2600. So a = 780. 

So box A now has 780 yellow ones and 520 orange ones. 

Subtracting, we find that 310 yellow M&M's and 40 orange M&M's were moved. 

First you can subtract the parts Mr. Rand cut off which is \(18 - 1\frac{15}{16} - 3\frac{1}{8} - 4\frac{1}{2} = 8\frac{7}{16}\) pounds left because 16 ounces make up a pound. After dividing it by \(\frac{9}{16}\) we get \(\frac{135}{16} \div \frac{9}{16} = \frac{135}{16} \times \frac{16}{9} = 15\) patties. Dividing 15 by 3 we get 5 packs of 3 patties for $3.12, so \(3.12 \times 5 = \boxed{\$15.60}\)

By the way this is what i did, and its wrong, please help still

real solution:

$(-(7 π)/4 + n π)$ in radians

$*180/π$

$-315+180n$

$-315+180n=0$

$n=7/4$

$-315+180n=360$

$n=15/4$

$=1170$

Nice diagram

Let x be total candy

White candy = (3/4)x

Pink candy = (4/7)(x-(3/4)x)

= (4/7)(x/4)= x/7

Remaining green candy = x-{(3/4) +(1/7)}x

= (3/28)x

Also green candy =9

(3/28)x = 9

X= 84 candy.

Therefore total no. Of candy = 84

Let x be the total number of hearts in the  bag
 Then according to question
 
 Number white hearts =( 3/4)x=3x/4
 Remaining (pink+green)  hearts in bag
 x-(3x/4)= x/4
  Number of pink heart = (x/4)×(4/7)= x/7
 Number of green hearts 💚= (x/4)-(x/7)=9
⇒ (7x-4x)/28= 9
⇒3x=9×28
⇒ x=(9×28)/3 =3×28=84 hearts Ans.

WXYZ is a parallelogram where angle W is half of a right angle WX = 10, and XY = 144, What is the area of the parallelogram.

Angle W = 45º         WX = 10          XY = 144

A = 144(10 * sin45º)

Let Peter, James and Samuel have X, Y and Z numbers of stamps.

According to question,

X = Z/3

Z = 2Y

Z = 30 + Y

Now,

2Y = 30 + Y

Y = 30

Therefore, 

Z = 30+30 = 60

X = 60/3 = 20

Together they have 

X + Y + Z = 20+60+30 = 110

3y  - 2x =   7

3y  = 2x   + 7

y =  (2/3)x  +  7/3

Slope =  2/3

Slope of a  perpendicular line =  negative reciprocal  =    -3/2  = m

point     -5,-1

Prob of first red    =    3  out of 11        3/11

 Prob next is re     =    2 out of 10         1/5

3/11 * 1/ 5 =  3/55

tysm!!! that helped a lot :))

Number  of possible  sets  of  vowels =  C(5,1)   = 5

Number of possible sets of  consonants  = C( 21, 2)   = 210

So......   5 * 210   =     1050    choices

Let  x  =  0

Then   y  =   -5     and  5

Let y =  0

Then x =  -10   and 10

So  we  have  these  4 vertex points in clockwise order     A =  ( -10,0)   B =  (0,5)   C =   (10,0)  D = ( 0, -5)

The  area of triangle  ABC  =   (1/2)   base of 20  * height  = 5   =  (1/2) (20) (5)   = 50

And  because of symmetry.....the area of triangle ADC is  the  same

So.....the  total  area  =      2 (50)  =    100

Hope that makes sense   !!!!

Point slope form -> y - 0 = 3/7(x - (-14))

Simplifying gives y = (3/7)x + 6

y =  2x  - 6   =  2 ( x - 3)

y = x^3  - 2x^2  - 3x  =  x ( x - 3)  ( x +1)

Intersection points

x ( x - 3) (x + 1)  =  2 (x - 3)

x ( x - 3) (x + 1)  - 2(x - 3)  =   0

(x - 3) (x^2 + x  - 2)   = 0

(x - 3) ( x + 2) ( x - 1)  =  0

We  have   three ontersection pts      x =  -2      x =  1       and  x =  3

See  the graph here  :  https://www.desmos.com/calculator/1l3ht5pza6

We have two areas to consider

1                                                                 3

∫   (x^3  -2x^2  - 3x)  - (2x - 6)   dx       +     ∫   (2x - 6)  -  (x^3  - 2x^2  - 3x)  dx      =

-2                                                                1

                1                        1                  1         1         1             3        3                  3

[  (1/4)x^4 ]       -  [(2/3)x^3]  - [(3/2)x^2 ]  - [x^2] +  [6x]  +  [ x^2]  - [6x]   - [ (1/4)x^4]      +

                -2                    -2                  -2         -2       -2            1         1                  1

               3                        3

[ ((2/3)x^3]    +  [ (3/2)x^2 ]         =

               1                       1 

-15/4    -6   + 9/2   +  3  + 18  + 8  -12   -20  +  52/3   + 12        =    253  / 12     

x^2 + 3x >= -2

x^2 + 3x + 2 >= 0

(x + 1)(x + 2) >= 0

Solution: [-2,-1]

its 32