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OK, DC...we have this

3.90625  =  500 (1/2) ^(t/5.23)      where t is in days

Divide both sides by  500

3.90625 / 500   =  (1/2)^(t/5.23)     take the log of both sides

log  ( 3.90625 / 500)  =  log (1/2)^(t/5.23)   and we can write

log (3.90625 / 500)  =  (t / 5.23) * log (1/2)

Multiply both sides by    5.23 / log (1/2)

log (3.90625 / 500) * (5.23 / log (1/2) )  =  t  ≈ 36.61 days

To find the number of half-lives  we have     

36.61 / 5.23   =  7 half-lives

x^2 + 4x  - 10  =  8x - 14

Subtract  8x , add 14  from / to   both sides

x^2 - 4x + 4  =  0       factor

(x - 2) ( x - 2)  =  0

Setting either factor to 0 and solving for x   prduces  x  = 2

And  using  8x  - 14.....  y  =  8(2) - 14  =  16   -  14   = 2

So the solution is (2,2)

Again....we can set the equations  equal

x^2  - 5x + 6  =  -4x + 6       add 4x, subtract 6   to / from each side

x^2 - x = 0     factor

x ( x - 1)  = 0

Setting each  factor to 0   and solving for x produces x = 0  or x  = 1

And using    -4x  +  6

When x  = 0 , y  = 6

When x  = 1, y = 2

So...the solutions are  (0,6)  and (1,2)

1.

Since y equals y...we can set the frst two equations equal

x^2  - 2x + 4   =  -2x + 4       subtract  -2x + 4 from both sides

x^2  =  0    ⇒    x  =  0

So...only one solution  at  (0, 4)

Here's a graph   : https://www.desmos.com/calculator/pmhadfbhr1

Didn't know that you were fluent in Spanish!!!!....LOL!!!!

Note that

Floor   (  8  < h  < 9)  =  8

and

Ceiling  ( 8 < h < 9)   =  9

So

Floor ( 8 < h < 9)  *  Ceiling ( 8 < h < 9)  = 

8 * 9   = 

72

The interval   is     8 < h < 9

Gracias

Thank you for all of your help, I really appreciate it!

Nice, hectictar   !!!!

Note 

sin 23  =  cos(90 - 23)   =  cos (67°)....so x  = 67

The second one seems trickier than it really is

RP....note that

sin P  =  cosR       and

cos P  =  sin R

so

Sin R  - cos R   =

Cos P  - Sin P  =   .8192   - . 5736  =   0.2456

Edit to correct a previous mistake  !!!

volume of candle   =   \(\frac13\pi r^2h\)          Since  h = r , we can replace  h  with  r .

volume of candle   =   \(\frac13\pi r^2r\)

volume of candle   =   \(\frac13\pi r^3\)

surface area of candle   =   \(\pi r^2+\pi r\sqrt{r^2+h^2}\)        Since  h = r , we can replace  h  with  r .

surface area of candle   =   \(\pi r^2+\pi r\sqrt{r^2+r^2}\)

surface area of candle   =   \(\pi r^2+\pi r\sqrt{2r^2}\)

surface area of candle   =   \(\pi r^2+\pi r\cdot r\sqrt{2}\)

surface area of candle   =   \(\pi r^2+\pi r^2\sqrt{2}\)

ratio of the volume of candle to its surface area  =   \(\frac{\text{volume of candle}}{\text{surface area of candle}}\)

\(\frac{\text{volume of candle}}{\text{surface area of candle}}\,=\,\frac{\frac13\pi r^3}{\pi r^2+\pi r^2\sqrt2}\\~\\ =\,\frac{\frac13r^3}{r^2+r^2\sqrt2} \\~\\ =\,\frac{\frac13r}{1+\sqrt2} \\~\\ =\,\frac{r}{3(1+\sqrt2)} \\~\\ =\,\frac{r(1-\sqrt2)}{3(1+\sqrt2)(1-\sqrt2)} \\~\\ =\,\frac{r(1-\sqrt2)}{3(1-2)} \\~\\ =\,\frac{r(1-\sqrt2)}{-3} \)

(4/x + 2)  /   ( (1/2)x  - 8 )  =

[ (4 + 2x) / x ]  /  [ (1/2) ( x - 16) ]   =

[ 2 (2) ( 2 + x) / x ]  / (x - 16)  =

[ 4(2 + x)]  / [ x ( x - 16) ]  =

[ 8 + 4x ]  / [ x^2 - 16x) ]

The line of reflection is  t .

Statements  1  and 2 only

These statements are the definitions of  sin (x) and cos(y), respectively

Statement 4  is a consequence of substitution

So  "C"  is correct

Thanks so much!

Correct...!!!

Due to symmetry  the x coordinate of the vertex = 5

The points  (3,0) (7,0) and (2, - 5)  are on the graph

And we have this form

y  =  ax^2 + bx + c

The x coordinate of the vertex  =  

-b / (2a)  =  5

-b  = 10a

b  =  -10a

So we have that using  (3,0)

0  =  a(3)^2 - 10a(3)  + c

0 =  9a - 30a + c

0  =  - 21a + c

-c = -21a

c = 21a

And using  (2, -5)

-5  =  a(2)^2  - 10a(2) + 21a

-5  = 4a - 20a + 21a

-5  = 5a

a  =  -1

And b =  -10(-1)  =  10

And c = 21(-1)  = -21

So our function is :

y = -1x^2 + 10x  - 21

Here;s the graph : https://www.desmos.com/calculator/u2qnb8dswx

revenue  =  R(h)  =  20 + 30h

cost  =  C(h)  =  10h² - 80

In how many hours does she break even?  This means...

How many hours does it take for revenue to equal cost?

What is the value of  h  when  revenue  =  cost   ?

What is the value of  h  when   20 + 30h   =   10h² - 80   ?

20 + 30h   =   10h² - 80      We want to solve this equation for  h .

                                           Subtract  30h  from both sides.

20   =   10h² - 30h - 80

                                           Subtract  20  from both sides.

0   =   10h² - 30h - 100

                                           Divide through by  10 .

0   =   h² - 3h - 10

                                           Factor the right side of the equation.

0   =   (h - 5)(h + 2)

                                           Set each factor equal to zero.

h - 5  =  0    or    h + 2 = 0

h  =  5         or    h  =  -2

Since  h  is the number of hours it takes to break even, we only want to include the positive solution.

h  =  5

It takes  5  hours to break even.

So it's C

Remember that....if V and T are complementary....the sine of one angle  = the cosine of the other

So......

cos T  =  sin V

cos T  = 0.84     and   sin V  = 0.84

First one

A tangent to a circle meets a radius at right angles

So  ABD  =  90°

Second one :

Ths is the intersecting chords theorem....specifically, we have that

9(n + 7)  = 18(n + 1)    divide both sides by 9

n + 7  =  2(n + 1)     simplify

n + 7  =  2n + 2   subtract n, 2  from both sides

n   = 5

Last one

Notice that ABC  =  180  - 47 - 48  =  85°

But ABC  would have to  = 90°   for BC to be tangent

So...the second  answer is correct

Correct !!!

4.

x + 2               x + 4   

____     ÷       _________           factor the denominator on the right

x  - 1              x^2 +4x - 5

x + 2              x + 4

____   ÷         __________         notice that x  cannot be  1  or  -5

x - 1              (x - 1) ( x + 5)  

So....we actually have

x + 2             (x - 1) (x + 5)

_____    x   ____________     =

x  -  1              x + 4

( x + 2) ( x - 1) ( x - 5)

__________________          the  (x -1)'s   cancel and we're left with

  (x - 1) ( x + 4)

( x + 2)( x - 5)

___________       Notice that we have another restriction......x cannot  = -4

      x +  4

So....the final answer is in red...and the restrictions are that x  cannot equal  1,-5, -4

So.....last answer

Remember that

sin x  =  cos(90 - x)

So   cos (90 - x)   =  0.84

Yes, it does make sense. Thank you!