geno3141

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 ...

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

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!