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We can use the Euclidean Algorithm. \(\begin{align*} &\text{gcd}\,(a^2+9a+24,a+4) \\ &\qquad=\text{gcd}\,(a^2+9a+24-(a+5)(a+4),a+4)\\ &\qquad=\text{gcd}\,(a^2+9a+24-(a^2+9a+20),a+4)\\ &\qquad=\text{gcd}\,(4,a+4). \end{align*}\) Since \(4\) is a factor of \(a\) and thus \(a+4\) , the greatest common divisor is \(\boxed{4}\).

Correct, Melody...!!!

f(x)   =  18  /  [ 6 + 5e^-0.1x ]

So

f(6)  =    18 / [ 6 + 5e^{-0.1 * 6} ]  =   18 /  [  6 + 5e^-0.6 ]   ≈  2.1

3.

y  =  (5/6) (x + 4) + 3   ⇒  (5/6)x + 10/3 + 3  ⇒  (5/6)x + 19/3 

y  =  3

y = (2/7) (x - 5) + 6  ⇒  (2/7)x - 10/7 + 42/7  ⇒  (2/7)x + 32/7 

So....a possible piece-wise function is

               (5/6)x + 19/3         if   -10 ≤ x ≤ -4

y  =            3                         if     -4 < x  ≤ 5

                (2/7)x  + 32/7        if   5 < x  ≤ 12

So 2 minutes past 12 does not count because the hand has gone fractionally past the 12?

f^-1  does exist

 f^{-1}(f^{-1}(f^{-1}(1)))  =

(  f^-1  °    f ^-1   °   f^-1 ) (1)  =

( f^-1 °  f ^-1) (2)  =

f^-1 ( 5)  =

3

1.    Range  =  [ -10, 0 )  U  [6, 13 ]

2.    (−5, 33) and (2, 5)

Slope  =   (33 - 5)  / ( -5 - 2)  =   28/ -7  =  - 4

Equation

y  = -4 ( x - -5) + 33

y  =  -4x - 20 + 33

y  =  -4x  + 13

4.   g(x)  is f(x)  shifted 3 units to the left and 6 units up  ...so...

g(x)    =   f(x + 3)  +  6

Since 1428 = 2^2 * 3 * 7 * 17, and the first factor, or 4, is common to the polynomial

[a^2 + 9 a + 24 and  a + 4], then no matter what value "a" takes, the GCD of a*1428 and the polynomial will always be the same, namely: 2 x 2 = 4. 

The diameter will be the longest side of the triangle......If we let this   = 2r, the height of the triangle is r

So.... the area of the triangle  is  

(1/2)(2r) (r)   =  r^2

And the area of the circle is pi * r^2

So.....the ratio of the area is  

r^2  / [ pi * r^2 ]  =

1 / pi    ≈   31.8%

That is really cool.

Thanks   XSquaredFactor   

We are looking for the values that make  f(x)  =   8

So we have

x^2 - 7x + 18  =  8       subtract 8 from both sides

x^2 - 7x  +  10  =  0      factor

(x - 5) ( x - 2)  = 0

Setting each factor to 0 and solving for x we have that  

x  = 5  or   x  =  2

So.....g(8)  =  either     2(5) + 3   =  13    or

                                    2(2)  + 3   =  7

And the sum of these  =   20

We have 

(2/5) * (Prob 2nd draw red   )   =  1/7       multiply both sides by  5/2

(Prob second draw red   )   =   1/7  * 5/2  

(Prob second draw red    )   =   5 / 14

Hypotenuse  = 

√  [ ( leg 1)^2  + (leg 2)^2  ]  =

√ [  12^2  +  9^2  ] =

√  [ 144 + 81 ]  =

√225  =

15

By the given information, we know the following:

\(t(x)=\sqrt{3x+1}\\ f(x)=5-t(x)\)

In order to find \(t(f(5))\), we must first evaluate f(x) when x=5:

Now we know that \(t(f(5)=t(1)\) since we just determined that \(f(5)=1\).

Therefore, \(t(f(5))=2\)

5.

a.  Minimum height  =  .5        Max height  =  4

b.  It will reach a minimum after 1 second

c.  Height after  2.5 seconds  =  

Note : cos (2.5 pi)  =  same as cos (pi/2)  =  0

1.75cos (2.5 pi ) + 2.25  =  1.75 (0) + 2.25  =   2.25

Here's a graph : https://www.desmos.com/calculator/lcxviz3ffb 

Go online to this page which explains the "sample mean" and "population mean" very well:

http://www.differencebetween.net/science/difference-between-sample-mean-and-population-mean/

Let x  =  0.....then  y  = 5(0)  = 0

So  (0,0) is on the graph

Now......let x  = 1.....then y  = 5(1)  = 5 

So   (1, 5)  is on the graph

Plot   (0,0)   and  (1, 5)....then draw a line through them.....this will be the graph 

It should look like this  :  https://www.desmos.com/calculator/ettknmo0vy

P ( A I B )  =  P ( A ) ???

P( A I B )  =   P (A and B ) / P (B)  =  .24 / .90  =  4/15   ≈  .267  

And P(A)  = .3

So....not independent because P(A|B) ≠ P(A)   [ the first answer ]

Yes, that is fine because 5/8 =0.625 x 100 =62.5%

Sum them up as geometric series:

Sum =F x (1 - R^N) / (1 - R), where F=First term, R=Common Ratio, N=Number of terms.

Sum =600 x (1 - 1.15^5) / (1 - 1.15)

Sum =600 x (1 - 2.0113571875) / -0.15

Sum =600 x (-1.0113571875) / -0.15)

Sum =600 x        6.74238125

Sum =$4,045.43

So I found that 5/8 works for the answer but I had to put it like this 62.5 %.

But your question says "What is the probability of getting anything but one head and two tails?"

So, the probability for ANY OTHER combination would be:

1 - 3/8 =5/8 !!!

That's what I tried to type in for my answer it was wrong. I tried .375 and then I tried it rounded off 38 % both were wrong.

I know the first two....unsure about the third

Not Independent   ..."Without replacement" usually always signals this

Independent  ...  the outcome of one cube does not affect the outcome any other  

Outcomes of tossing three coins

T  T   T

T  T   H

T  H  H

T  H  T

H  H  T

H  T  H

H  T  T

H  H  H

Prob  of  one head and two tails   =  3 / 8