CPhill

B.Triangle DEC is congruent to triangle DBC.

We have this possibility

f(x)  = - (x+ 4)^2 (x - 0) ( x - 6)  =   -x^4 - 2 x^3 + 32 x^2 + 96 x

A zero of an odd multi[licity will cut through the x axis

A zero with an even multiplicity will just "kiss" the x axis

Since the graph goes to -infinity  as x approaches both infinities, we need the  " -"  in front of the first term.......the end behavior of this term will overpower the end behavior of all the other terms

Heres   a graph to get a feel for this :

First one :  Either method is correct

The first student is evaluating  [25^(1/2)]^3    while the second student is evaluating (25^3)^(1/2)

Both lead to the same result

Second one

h  =  - 16t^2 + 50t + 6....we want to find the time when the height of the ball = 41 feet

So.... we can solve this

-16t^2  + 50t + 6   =  41    subtract 41 from both sides

-16t^2 + 50t - 35 =  0     multiply through by -1

16t^2 - 50t + 35  =   0

Using  the quadratic formula   where a = 16, b  = -50  , c  = 35

x  = ( - b  ±√[ b^2 - 4ac] )  / 2a  =   ( 50 ±√ [ (-50)^2 - 4(16)(35) ] ) / (2*16)  =

(50 ±√[ 260]) / 32  

Taking  x  =  ( 50 + √[260] ) / 32

x  = 2.07 sec

f(x)  = (x - 3)^3  - 4x^3 (x + 2)^3 + 3(x - 1)^6      expand this

f(x)  = -x^6 - 42 x^5 - 3 x^4 - 91 x^3 + 36 x^2 + 9 x - 24

The end behavior will be determined by the first term...its end behavior "overpowers" the end behavior of the other terms

As x  approaches negative and positive infinity, f(x) will approach negative infinity

So "A"  is correct

Look at the graph here to get a feel for this :

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

\( \frac{x + 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\)

We can use partial fraction decmposition to find A, B

Factoring the denominator on the left side  we have   (x - 2) ( x + 1)

Multiply through by this common denominator  and we have

x + 7  =  A(x + 1)  + B(x - 2)   simplify

x + 7  =  Ax + A  + BX  - 2B        equate coefficients  and we get this system

A + B  = 1

A - 2B  = 7       subtract the second equation from the first

3B  = -6     divide both sides  by 3

B  = -2

Using the first equation to  find A

A +  -2   = 1         add 2 to  both sides

A  = 3

So  (A, B )   =  ( 3, -2)

59

We have this relationship :

PS / SR   = SR / SQ      ....so...

4 / SR  = SR / 16      cross-multiply

4 * 16  =  SR * SR

64  = SR^2       take the square root of both sides

8 = SR

60

We can us the Pythagorean Theorem to  find PR

PR  = √[PS^2  + SR^2 ] =  √[ 4^2 + 8^2 ] = √[16 + 64 ]  = √ 80  =   √[16 * 5 ] =  4√5  units ≈ 8.9 units

Applying the sample data.....25   wanted cheese pizza...of these....13 wanted stuffed crust....so

13  / 25  =      52  / 100    =    52%

G = (7,3)    H = ( 9,0)    I  = ( 5, -1)

Distance from   G to H  = √ [ (9- 7)^2 + (3 - 0)^2  ]  = √ [ 2^2  + 3^2  ]  =  √13

Distance  from G to I  = √ [ (7 -5)^2 + (-1 - 3)^2 ]  = √ [ 2^2 + 4^2]  = √20

Ditance  from H to I  =  √ [ (9 - 5)^2  + 1^2  ] =  √ 17

Defintely  NOT an equilateral triangle......nor  a right triangle   because GH^2 + HI^2  > GI^2

We have  an acute triangle......if it was obtuse   GH^2 + HI^2   would be  <  GI^2

See the pic here to verify this :

Don't know much about this but I think I can  feel  my way around......LOL!!!

Mean  = $240

SD  = $100

Joe's Balance = $260

Z  =   [ Joe's Balance  - Mean Balance ]   /  SD  =

[ 260  - 240 ] / 100   = 

20/100  =  

.2

Looking up .2 in  a  table of Z scores we find that  the value is  .5793

So....58%  have a balance  less than Joe's

50.   We just multiply each coordinate by the dilation (scale) factor to get the new image ......

A  = ( -6, - 2 )    after dilation of  1/2  =      A'  = (-3, -1)    after dilation of A ' by 5  =  (-15, -5)

B  = (-2 2)   after dialation of 1/2  = B'   = (-1.1)  after dialtion of B' by 5  = (-5,5)

C = (4, -8)  after dilation of 1/2  = C'  = (2, -4)  after dilation of C' by 5  = (10, -20)

51.....pretty easy, RP  !!!!

We just multiply the the original coordinates by  the product of both scale factors=  (1/2) * 5   = 5/2

Note that this  produces the following

5/2 * (-6, -2)   =  (-15, -5)

5/2 *  ( -2. 2)  =  ( -5, 5)

5/2 * ( 4, -8) =  (10, -20)

Verify that these are exactly the same as produced in  50  !!!!