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Well, heck....anytime!

Dang!   Wish I was 22 again !      Haha .....

I've answered a lot of chemistry Q's also......ask away !      ...or  Physics.....     etc

32.

 numbers greater than 4, or prime numbers.

2,3,5,6

4/6

=2/3

19. Since the pattern is times 3 from the previous number, and we want to find the tenth number, here's the list:

1+3+9+27+81+243+729+2187+6561+19683

Adding gives us:

29524

Actually you helpped me out a lot by helping me get to what my teacher was asking for. :)!!

25.

x^2    +    y^2         

___        ___      =        1

49           25

26.

The distance between  the focal points  is  given by :

2 * √ [ 10^2  - 8^2 ]  =

2 * √ [ 100  - 64 ]  =

2 * √36  =

2 * 6  = 

12 ft

21.   

This is an ellipse....in standard form, we have

x^2   +  y^2      =     1

___     ____

  1         4

The domain  is    -1  ≤ x   ≤ 1

The range  is  -2  ≤ y  ≤ 2

22.

Vertex =  (0,0)

Focus  (-4,0)

So we have

4px = -y^2         

p  = 4.......so we have

4(4)x  = -y^2

16x  = -y^2 

x  = (-1/16)y^2 

31.

Area  of the  board  =  pi * 16^2  = 256 pi

Area  of the bulls-eye  = pi *  6^2  =  36 pi

So....the probability  is     36  / 256  =   9/64  ≈  14.1%

32.

P ( prime  or > 4)   =  P(prime) + P (> 4)  - P (prime and  > 4)  =

                                   3/6           +  2/6    -  1/6   =

                                              4/6  =

                                              2/3

29.

We want to permute   all possible orders of selecting any 3 students from 10....so we have

P (10,3)   = 720 ways

30.

We have 36 possible outcomes.....the ones that sum to 6  are

1,5

5,1

4,2

2,4

3,3    =     5 ways

So...the probabilitiy  is     5 / 36

3. lim x -> 1 f(x) if f(x) {x+2 if x = 1, 2x^2 - 1 if x >1

When x approaches  1 from the left, f(x)  = (1) + 2  = 3

When x  approaches 1 from the right, f(x)  = 2(1)^2  - 1  =  1

So.....this limit  does not exist as  x approaches 1 because both limits would have to be the same approaching from the left and the right

Let's see....I'm going to use  "n"  instead of "k" to prevent confusion

Cross product

       i             j         k           i          j

   n + 1         1         -2      n + 1      1

     n             3          0         n         3

[ 0i  +  (-2n) j  + (3n + 3) k ] -  [ n k  - 6i  + 0j ] =

6i  + ( - 2n)  j + (2n + 3)k  

So.....we have  ( 6 , -2n , 3 + 2n )

So...we wnat to solve this

√ [ 6^2  + (-2n)^2  + (3 + 2n)^2  ]  = √41       square both sides

6^2  + (-2n)^2 + (3 + 2n)^2  = 41     simplify

36  + 4n^2  +  4n^2 + 12n + 9  = 41

8n^2  + 12n  + 45  = 41

8n^2 + 12n + 4  = 0

2n^2 + 3n + 1  = 0

(2n + 1) ( n + 1)  =  0

Setting  each  factor to 0   and  solving for  n we get that  n  = -1/2  or  n  = -1

So...converting back to k....then  k = -1/2  or k  = -1

MEMEMEM

2. lim x -> 0 

sin ( 1/x  )

This limit does not exist  ....see  the graph here :

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

1. Iim x -> 4

(1)/(x-4)

This limit will not exist.....as x  approaches 4 from the left  the function becomes increasingly more negative......as  x  approaches 4 from the right, the function becomes increasingly more positive

Here's a graph to show this  :

https://www.desmos.com/calculator/sxf0xscp3a

The slope of the tangent line at  x  = 1  is   16(1) - 4   = 12

And at x  = 1, then  y  =  8(1) ^2  -4(1)  + 3   =  8 - 4 + 3  = 7

So...the equation of the tangent line is

y  = 12 (x - 1)  + 7

y = 12x  - 12 +  7

y  =12x - 5

Here's the graph : https://www.desmos.com/calculator/daj2o31drm

The horizontal tangent line will be tangent to the vertex.......it will have a slope of  0

So...we have that the x coordinate of the vertex is

16x - 4  = 0

16x  = 4

x  = 1/4

And when x  = 1/4, y  =  8(1/4)^2 - 4(1/4) + 3  =  1/2 - 1 + 3  =  2.5  = 5/2

So.....there is a horizontal tangent line at  ( 1/4, 5/2 )

We can make 7 different shakes if we use the same flavor

If we use different flavors, we can make   C (7 ,2)  =  21 diffferent shakes

So.......7  +  21   = 28 different shakes can be made

Using the Law of Sines

sin 69 / 35  = sin R / 30    multiply  both sides by 30

(30/35)sin69  = sin R

(6/7) sin 69   = sin R

Use the sine inverse to find R

arcsin [  (6/7) sin 69 ]  =  R  = 53°

We can use the Law of Sines,here

sin 61 / 45  = sin C  / 35      multiply both sides by 35

35 * sin 61 / 45  = sin C

(35 / 45) sin 61  = sin C

(7/9)sin 61  = sin C

Use the sine inverse to  find  C

arcsin  [ (7/9) sin 61 ]  = C  ≈   42.86° = 43°

3)  I modified your equation: e^-(xk)

L(x)=100e^-((k)x)

L(9) =100*e^-(0.2*9)

L(9) =100 * e^-1.8

L(9) =100 * 0.1653

L(9) =16.53% - percentage of sunlight that penetrates water at a depth of 9m.

5% = 100 * e^-(0.2x),        solve for x

0.05 =100 * e^-(0.2x)        divide both sides by 100

0.0005 = e^-(0.2x)             take the natural log of both sides

-0.2x = -7.6                        divide both sides by -0.2

x = 38 meters - at which only 5% of sunlight will penetrate the water.

Hello Guest!

This is an equation of a line.

We can rewrite it in point-slope form:

-2x = -1 - y

2x=1+y

y=2x-1

We can then graph this line by plugging in 2 points then connecting the two points. 

Two points that work are:

(0, -1) and (2, 3) 

You can then connect those two points on a graph and all the points on the line are solutions. 

I hope this helped,

Gavin

Point-Slope form is:

y - b = m(x - a)

Where m is the slope, and (a,b) is a point the line passes through. 

Since we are already given the points, we can plug them in to update our equation:

y - 7 = m(x - (-5)) 

Simplyfied:

y - 7 = m(x + 5) 

We now need to solve for the slope. 

The slope is rise over run. 

Rise is the first y value minus the second y value. 

Run is the first x value minus the second x value

Rise is 7 - 0 = 7 

Run is -5 - ( -4 ) = -1 

The slope is -7. 

The equation in point-slope form is y - 7 = -7(x + 5) 

Thank you SO much for all the answers Omi!  These were so helpful!

4)

140 = 200 * (1/2)^(3/h), where h = half-life.

Divide both sides by 200

0.7 = 1/2^(3/h) 

Take the natural log of both sides

3/h =Ln(0.7) / Ln(1/2)

3/h =0.514573173

Cross multiply

0.514573173h = 3

Divide both sides by 0.514573173

h =5.83  hours - the half-life of this radioactive substance. 

1.5 days x 24 hours = 36 hours.

Use the same above formula:

R =200 * 1/2^(36/5.83)

R =200 * 1/2^(6.175)

R =200 *   0.0138401175...

R =2.768 grams will remain after 1.5 days or 36 hours.

Seven-S=even