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The two triangles, triangle(APB) and triangle(CPD) are similar by AA.

The corresponding lengths of sides of similar figures are in the ratio of the square roots of the areas.

So:  AP / PC  =  sqrt(4) / sqrt(9)   --->   AP / PC = 2 / 3.

Let's call the length of AP = 2x, which makes the length PC = 3x; also, BP = 2x and PD = 3x.

A formula for the area of a triangle is:  A  =  ½·a·b·sin(C).

Using this formula on triangle(APB):  a = AP = 2x     b = BP = 2x     A = 4 cm² 

     --->     A  =  ½·a·b·sin(angle( APB) )     --->     4  =  ½·(2x)·(2x)·sin(angle( APB )     

     --->     sin( angle(APB) )  =  2 / x² 

Similarly, sin( angle(CPD)  =  2 / x² 

Area of triangle(APD)  =   ½·(2x)·(3x)·sin(angle( APB )  =  (3x²)·sin(angle( APB )

(since sin( angle(CPD)  =  2 / x² )     --->     Area of triangle(APD)  =  (3x²)·(2 / x²)  =  6 cm²

Therefore, the total area will be  4 cm² + 9 cm² + 6 cm² + 6 cm²  =  25 cm² 

Let the number of berries in the bowl =B
Billy =1/2B + 2
Bob =1/3*1/2B + 2 =1/6B + 2
Joe =1/3*1/6B - 1=1/18B - 1
 [1/2B+2+1/6B+2+1/18B] =B - 1, solve for B
B = 18 Berries originally in the bowl.
Billy took =9+2=11
Bob took =3+2  =5
Joe took =1 and left 1 in the bowl

We can use slant height this way:  V = [ 12 * 12 + (6√3) ] / √13.5 = 407.293506 u³   

\(y = ax + b \) is our general form for this line.

We know the points (4,5) and (8,17) MUST satisfy this equation as they are part of this line. So, we can substitute this into the equation.

1) \(y = ax + b \)

    \(5 = 4a + b\)

Also,

2) \(y = ax + b \)

   \(17 = 8a + b\)

We can now solve this system of equations!

We subtract equation 1 from equation 2.

      \(17 = 8a + b\)

(\(-\)) \(5 = 4a + b\)

     \(12 = 4a\), isolate a by dividing by 4 on both sides.

     \(a = 3\)

Now, we can find be by substituting the value of a into either equation.

   \(5 = 4a + b\)

\(​​​​5 = 4(3) + b\), subtract 12 from both sides.

   \(b = -7\)

However, this is not our answer! MAKE SURE TO ALWAYS ANSWER THE QUESTION!!

\(a - b = 3 - (-7) = 10\)

So, \(a - b = 10\)

Hope this helps Guest!

nvm i figured out the question

the answer is \(\sqrt{2}\) and your hint was actually really helpful thx!

Hello, Guest!

" height = 6 root 3 "     this is slant height

You need the altitude (height) of the pyramid to calculate the volume.

I'll use your slant height to find the height of the pyramid.

h = sqrt[( 6*√3)² - 6² ] = sqrt( 108 - 36 ) = 8.485281374  

\(4 \frac{3}{10} + 3 \frac{2}{5}+ \frac{1}{3} + \frac{1}{2}=\boxed{\frac{128}{15}} \text{ or } \boxed{8\frac{8}{15}} \)

1)   Diagonal of a square   d = sqrt( 12² + 12² ) = 16.97056275            d/2 = 8.485281374

2)   Height of a pyramid     h = sqrt( 12² - 8.485281374² ) = 8.485281374 ()

3)   Volume of the pyramid     V = lwh/3 = (12*12*8.485281374) /3 = 407.293506 u³  

hint:  check if this triangle is the right or "wrong" triangle.

 the fewest number of toothpicks that could be removed is 10  

thats wrong

so we have sqrt(10^2+10^2) which is sqrt200 so the area of the square is 200 and then we do 10*10/2=50 so the area is 200-60=150

ok thanks

this is the definition of cos

In a right angled triangle, the cosine of an angle is: The length of the adjacent side divided by the length of the hypotenuse. The abbreviation is cos. cos(θ) = adjacent / hypotenuse.

so we get (2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9)/(x^2 + 2x + 3) becomes 2x^3-x^2+4x+3

-6 since the equation is -3x+-9y=3 see graph here:

https://www.geogebra.org/calculator/vywdwnmd

356  365  536  563  635  653  Total =  6

Probability = 6 / 6^3 =6 / 216 =1 / 36

It would converge to 5, no matter what number you started with!! I wrote a little code starting with number 37.This is the result after 5 iterations:

i=0;a=37;b=20; c=2# (d+b);d=2#(c + b);printd;i++;if(i<10, goto3, 0)

5.002481161
5.000024811
5.000000248
5.000000002
5
 

nvm i think i figured it out

It can be \(-\sqrt\infty+\sqrt{\infty+1} \text{ And that will become } \sqrt{4\infty+1}\) where \(\infty \text{ stands for the largest number}\). 

15 since we have that the first slope is -3/4 and so the slope of line₂ is 4/3 and it passes through the point (-5,7) and we multiply that by 5 to get 20/3 and add 7 to get 41/3 then we add 4/3 and 41/3 to get 45/3=15

If the length of the hypotenuse of the right triangle equals the side length of the square, find the area of the yellow region.

yelow area is   10² * 2 - 10² / 2   

So for perpendicular bisectors has slope 1/6 since the first one has slope -6 and we then get thta the y-intercept is 3 1/3 so we get that the answer is 7/2.

So for perpendicular bisectors has slope 1/6 since the first one has slope -6 and we then get thta the y-intercept is 3 1/3 so we get that the answer is 7/2.