All Answers 358232 Answers

Ah....pretty crafty, lynx.....!!!!!!

545+5+5=555

Mmmmm...I only added  one straight line ....well, actually...only a very short segment....LOL!!!

there could be two answers well spotted cphill

Mr. Smith brings home 7 animals for his 7 children. Each child will adopt a pet to be her or his own. There are 4 different cats (a Siamese, a Persian, a Calico, and a Minx), 2 different dogs (a Poodle and a Golden Retriever), and a goldfish. Anna and Betty refuse to take care of the goldfish, and Charlie and Danny insist on having cats. The other 3 kids are easier to please -- they'll take anything. In how many ways can Mr. Smith give the children pets they'll want?

First we must make sense of this:

cat1, cat2, cat3, cat4,      Charlie and Danny and 2 others

Dog1, dog2,   

GoldFish                          NOT  Anna or Betty

I am going to attmept to allocat the pets in this order.

Charlie (4 cats), Danny (3 cat possibliities), Anna (4 non-fish pets left), Betty (3 non-fish left) , C5 (3 pets left), C6(2 pets left), C7 (this child gets what is left)

4*3*4*3*3*2*1 = 864 ways to allocate the pets.

it says that is incorrect cphill

It is the x that has changed. I mean x has been repleaced with 9.2x so it is a horizontal compression.

multiplying x by 9.2 compresses the graph by 1/9.2 = 5/46 

5/46 is a approximately 0.11 I tell you this so you can understand the graph. 

You can see that   (1,0) has compressed to (0.11,0)

Good try, lynx...but notice that it might also be true that

a + b  =  3

0 + 3  =  3

So........

0^5  + 3^5   =   243  !!!!

May be cheating....but....

5 + 5 + 5 + 5  ≠  555

a+b=3

1+2=3

a^5+b^5

1^5+2^5

1+32=33

a^5+b^5=32

32 must be the answer unless it is in decimals.

log₃ (4*3^{x- 1}  - 1)   = 2x - 1   write this in exponential form

3^(2x - 1)  =  4*3^(x - 1) - 1

3^(2x) / 3  =  4*3^x / 3  - 3/3       multiply through by 3

3^(2x)  = 4*3^x  -  3

(3^2)^x  = 4*3^x - 3

(3^x)^2  = 4*(3^x)  -  3

Let  a  =  3^x

a^2  =  4a  - 3

a^2  - 4a + 3   =  0       factor

(a - 3)  (a - 1)  =  0

Set  each factor to 0  and solve for a and we get that

a  =  3    or    a  =1        which means that

1 =  3^x    ⇒  x = 0      or  that

3  =  3^x   ⇒   x  =  1

Both solutions solve the original equation

I'm sorry...but I don'tknow what "domain of the equation"  means....unless we require that 

4*3^(x - 1)  -  1  >  0    if so.....then

4*3^(x - 1)  .  1

3^(x - 1)  > 1/4     take the log of both sides

(x - 1) log(3)  > log (1/4)

log (3) *  (x)  >  log (1/4) + log (3)

x > [ log (1/4) + log (3) ] / log (3)  

x >  log (3/4) / log(3)

x > ≈  -0.262  ⇒    this is the domain

Solve and find the domain of the equation:

\(\huge{ x^{\log_\sqrt{x} (x-2)}=9 }\)

1. Domain:

\(x-2 > 0 \\ x > 2 \)

\(\text{ $\{x \in R : x>2\}$ (assuming a function from reals to reals) } \)

2. Solve:

\(\begin{array}{|rcll|} \hline \large{ x^{\log_\sqrt{x} (x-2)} }& \large{=}& \large{9} \\ && \boxed{ \log_\sqrt{x} (x-2) = \dfrac{\ln(x-2)}{\ln(\sqrt{x})} \\ = \dfrac{\ln(x-2)}{\ln(x^{\frac12})}\\ = \dfrac{\ln(x-2)}{\frac12\ln(x)}\\ = \dfrac{2\ln(x-2)}{\ln(x)}} \\ \large{ x^{\dfrac{2\ln(x-2)}{\ln(x)}}}& \large{=}& \large{9} \\ && \boxed{\text{Formula: } a^b = e^{\ln(a^b)} = e^{b\ln(a)} } \\ && \boxed{ x^{\dfrac{2\ln(x-2)}{\ln(x)}} = e^{ \dfrac{2\ln(x-2)}{\ln(x)} \cdot \ln(x) } =e^{2\ln(x-2)} } \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline \large{ e^{2\ln(x-2)} } & \large{=} & \large{9} \\ \large{ e^{\ln\left((x-2)^2\right)} } & \large{=} & \large{9} \quad & | \quad e^{\ln\left((x-2)^2\right)} = (x-2)^2 \\ \large{ (x-2)^2 } & \large{=} & \large{9} \quad & | \quad \sqrt{} \text{ both sides} \\ \large{ x-2 } & \large{=} & \large{\pm 3} \\\\ x_1 -2 &=& 3 \\ x_1 &=& 3+2 \\ \mathbf{x_1} & \mathbf{=} & \mathbf{5} \\\\ x_2 -2 &=& -3 \\ x_2 &=& -3+2 \\ \mathbf{x_2} & \mathbf{=} & \mathbf{-1} \quad & | \quad \text{no solution, see domain } x> 2 \\ \hline \end{array}\)

First one

x^2 * log_x 27 * log₉ x  = x + 4      we can write

x^2  (log 27 / log x ) ( log x / log 9)  =  x + 4

x^2 (log 27 / log 9)  =  x + 4

x^2 ( log 3^3 / log3^2)  =  x + 4

x^2  *(3/2)  = x + 4

(3/2)x^2 - x - 4  = 0

3x^2 - 2x - 8   =  0     factor

(3x + 4) ( x - 2)  =  0

Setting each factor to 0   and solve for x  and we have that x = -4/3  or x  = 2

Only the second solution is good.....the first would mean that we were taking the log of a negative number in the original equation

Area of sector - Area of triangle  = Desired Area

The "formula" for the area is :

 r^2  [ (pi * 125/360)  -  (1/2) sin (125°)  ]  =

8^2 [  (pi * 125/360)  - (1/2)sin (125° ) ]  ≈  43.6  units^2

Thaks, helperid...here's an alternative solution

Without a loss of generailty we can let  AB  = 1

So   AHI  =   (1/2)(4)*2 sin (60)  =  4√3       (1)

Area  of  AFG  =  (1/2)(3)^2 sin (60) = (9/4)√3    (2)

So area of trapezoid  FGIH  =   √3 [ 4 - 9/4]  = √ 3 [ 7/4]

So  ratio of area of trapezoid FGIH  to AHI  =

√3 {7 /4]  / [  4√3 ]   =   7 / 16

EDIT TO CORRECT AN IDIOTIC ERROR  !!!!

Positive  on (-4, 0) and  (6,8) 

Increasing on  (-inf, 0)  and (1,2)

Constant on (-4, 0).....the function does not rise or fall in this interval

h² + 2h + 5   =   0

                                    First subtract  5  from both sides of the equation.

h² + 2h   =   0 - 5

Now we want to add a number to both sides that will cause the left side of the equation to be a perfect square trinomial. So add  (2/2)²  , which is  1 ,  to both sides of the equation.

h² + 2h + 1   =   0 - 5 + 1

h² + 2h + 1   =   -4

                                       Now that the left side is a perfect square trinomial, it factors like this...

(h + 1)²   =   -4

                                       And we can add  4  to both sides so the right side  =  0  again.

(h + 1)² + 4   =   0

The question didn't say to solve for  h , it just said convert to vertex form...I think that is it.

I'm sure I am not going to re-invent THAT wheel.!!   .... here is a place to get you started:

https://en.wikipedia.org/wiki/Poker_probability

I know absolutely NOTHING about playing cards, but go to Wolfram/Alpha and enter each hand and it will give you the probability such as "Royal Flush" for 7-card hand. See here:

https://www.wolframalpha.com/input/?i=52+playing+cards,++7-+card+Royal+Flush

Then just find the area of the sector and subtract the area of the triangle

sector area     =   125/360 x  pi x 8^2 = 69.813 sq units

Now subtract the triangle for the area of the segment.....

Sorry, could you please edit it so we could see what you are talking about?

Sorry if I didn't clarify. I wasn't looking for the area of the triangle, I was looking for the area of the part of the section of the circle that wasn't a part of the triangle, but was ont the other side of the chord with the sector.

Thanks though! 

The chord line and the the two  radii of 8  form a triangle of which one angle is 125 degrees and the other two are

27.5 degrees   (must add to 180 degrees)

  Draw a line of length 'h' perpindicular to the chord which bisects the 125 degree angle .

Now you have two equal triangles with angles of 90  125/2   and 27.5 degree angles

Now use law of sines to find 'h'  

sin 90 /8  = sin 27.5 /h = sin 62.5/(1/2chord)         

h = 3.694    and chord = 14.192

Then area of the triangle = 1/2 b h      where b is the chord

Area = 1/2 (14.192)(3.694) = 26.22 units^2