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The graphing form of the equation is:  x - h  =  a(y - k)²

where  (h, k)  =  (-4, 2)  is the vertex:  x - -4  =  a(y - 2)²   --->   x + 4  =  a(y - 2)²​  

To find the value of a, put in the coordinates of a point on the parabola -- I'm going to choose the point (-2, 0):

--->    x + 4  =  a(y - 2)²​    --->   -2 + 4  =  a(0 - 2)²​   --->   2  =  a(4)   --->   a  =  ½

The final equation is:   x + 4  =  ½·(y - 2)²​  

Multiplying this out:     x + 4  =  ½·(y² - 4y + 4)

                                    x + 4  =  ½·y² - 2y + 2

                                          x  =  ½·y² - 2y - 2

To find the diagonal of a prism, use the "Three dimensional Pythagorean Theorem":

Distance²  =  Length² + Width² + Height²

Formulas:  sin³(x)   =  ¾·sin(x) - ¼·sin(3x)      --->   sin³(2x)  =  ¾·sin(2x) - ¼·sin(6x)

                  cos³(x)  =  ¼·cos(3x) + ¾·cos(x)   --->   cos³(2x)  =  ¼·cos(6x) + ¾·cos(2x)

[ sin³(2x) ] · cos(6x)  + [ cos³(2x) ] · sin(6x)

=  [ ¾·sin(2x) - ¼·sin(6x) ] · cos(6x)  +  [ ¼·cos(6x) + ¾·cos(2x) ] · sin(6x)

=  ¾·sin(2x) · cos(6x) - ¼·sin(6x) · cos(6x)  +  ¼·cos(6x) · sin(6x) + ¾·cos(2x) · sin(6x)

=  ¾·sin(2x) · cos(6x) + ¾·cos(2x) · sin(6x)

=  ¾ [ sin(2x) · cos(6x) + cos(2x) · sin(6x) ]

But:  sin(A + B)  =  sin(A)·cos(B) + cos(A)·sin(B)

=  ¾ · sin( 2x + 6x )

=  ¾ · sin( 8x )

Your turn ...

Use cosine rule.

If you do not know what that is then google it.

It is a rectangle.

You must know some of those answers. 

Which ones do you know?

Maybe you need to be more specific with what is causing you problems........

Part a:  sin(x)·cos(x)  =  ½·sin(2x)

              h  =  ( v_o² / 16 ) · sin(x) · cos(x) 

--->        h  =  ( v_o² / 16 ) · ½ · sin(2x) 

--->      75  =  ( 50² / 16 ) · ½ · sin(2x)

--->   0.96  =  sin(2x)

--->      2x  =  sin^-1( 0.96 )

--->        x  =  0.6435^r

Part b:    h  =  ( v_o² / 16 ) · sin(x) · cos(x) 

--->     200  =  ( v_o² / 16 ) · sin(0.75) · cos(0.75) 

--->      v_o²  =  6416.072

--->        v₀  =  80.1  ft/sec

Wait, so if those are the edges do I just use the Pythagorean Theorem to find the diagonal?

Try these for the edges:  4 - 2sqrt(2),  4 + 2sqrt(2),  2.

The number of four-letter "words" (without any restrictions) is 5 x 5 x 5 x 5

The number of "words" withou any consonants is 2 x 2 x 2 x 2.

Subtract the second answer from the first to find the number of words that have at least one consonant.

Multiply the cost of the item by 1.06 to find the final cost.

There are 4 quarts in a gallon.

You need 3 gallons, which is 12 quarts.

Three-fifths of this must be blue and two-fifths of this must be white.

(3/5) x (12 qts)  =  7.2 quarts (which is almost 2 gallons)

(2/5) x (12 qts)  =  4.8 quarts (which is one gallon plus one quart).

Using these numbers, you can calculate the price.

To find AC, use the Law of Cosines:  AC² =  5² + 8² - 2·5·8·cos(20^o)

Then, you can find the size of angle(C) --

To find angle(C), use the Law of Sines:  sin(C) / 5  =  sin(20^o) / AC               

Then, you can find the size of angle(A) by subtracting angles B and C from 180^o.

[Warning!  Don't use the Law of Sines to find the largest angle in a triangle -- the answer that your calculator gives you may not be the correct answer!]

The angle is 36^o.

The opposite side is the height off the ground; call it x.

The distance to the tree is the adjacent side and is 14 feet.

tangent = opposite / adjacent

tan(36^o)  =  x / 14

tan(36^o) · 14  =  x                         <---  calculator time!

If you substitute 0 or x into these equations, you find that D is correct.

sec(x)  =  1 / cos(x)   --->   sec²(x)  =  1 / cos²(x)

csc(x)  =  1 / sin(x)   --->   csc²(x)  =  1 / sin²(x)   --->   1 / csc²(x)  =  sin²(x)

sec²(x) / csc²(x)  =  sin²(x) / cos²(x)  =  tan²(x)

List them ...

{5}

{1,5}     {2,5}     {3,5}     {4,5}

{1,2,5}     {1,3,5}     {1,4,5}     (2,3,5}     {2,4,5}     {3,4,5}

{1,2,3,5}     {1,2,4,5}     {1,3,4,5}     {2,3,4,5}

{1,2,3,4,5}

LIst them ...

1 - 2  =

1 - 3  =

2 - 3  =

2 - 1  =

3 - 1  =

3 - 2  =

Find the differences and count the number of different differences ...

cool, thanks!

yes that is what i mean , sorry! but okay great i got it right thank you for explaining~

AR and AT are both tangents to the circle; therefore, they are equal   --->   AT  =  ______

NE and NT are both tangents to the circle; therefore, they are equal   --->   NT  =  ______

GR and GE are both tangents to the circle; therefore, they are equal   --->   GE  =  ______

If you add all 6 portions of this triangle, you will get the perimeter.

Call the very top vertex of the pyramid A.

Call one of the corners of the pyramid B.

Call the midpoint of the base C.. 

Triangle(ABC) is a right triangle with angle(C) the right angle.

Side(AC) is the height of the right triangle.

Side(AB) is one of the edges of the pyramid and is the hypotenuse of triangle(ABC); it has length 12.

The first problem is to find the distance from C to B. 

This distance is one-half of the distance from B to its opposite vertex of the base.

Since the base of the pyramid is a square, each of its sides is 12. The distance from one corner of the square to its opposite corner is 12·sqrt(2).  [You can use the Pythagorean Theorem to find this length.]

So, the distance from C to B is one-half of 12·sqrt(2), which is 6·sqrt(2).

To find the height of the pyramid, we can use the Pythagorean Theorem on triangle(ABC).

[ 6·sqrt(2) ]² + [ CA ]²  =  [ 12 ]² 

            36·2 + [ CA ]²  =  144

               72 + [ CA ]²  =  144

                       [ CA ]²  =  72  

                            CA  =  sqrt(72)

                            CA  =  6·sqrt(2)     <---   height of the pyramid

The area of the base of the pyramid  =  12 · 12  =  144

Volume of the pyramid  =  (1/3) · Area of the Base · Height

                                      =  (1/3) · 144 · 6·sqrt(2)

                                      =  288 · 6·sqrt(2)

do you know the answer now. or how to get to the answer

Thank you 😊 

it's E

we know the whole way around the center is 360 degrees and since it's divided into 8 sections (not including the line that goes straight down) we can divide 360/8 to get 45 degrees for angle 1. Angle 2 is in the same triangle but just split in half, so since we already figured out what angle 1 is, we just split it in half to get angle 2. 45/2 = 22.5 degrees. Lastly, all triangles have to have their angles add up to 180 degrees so for number 3, we just figure out the other two angles in the small triangle that it's in. We know angle 2 is 22.5 and that the other angle is 90 degrees because the little square at the bottom of the octagon. That means for the little triangle that angle 3 is in, the angles are 22.5 degrees (angle 2), 90 degrees (given by the picture), and angle 3 (which we're trying to find) and it all adds up to 180 degrees. So the final step is taking the 180 degrees and subtracting 22.5 and 90 to get that angle 3 is 67.5 degrees.