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If a, b, and c are integers satisfying , , and abc = 21, then find . Express your answer as a common fraction.

\(\{a,b,c\}\subset \{1,3,7\}\ |abc=21\)

  !

Sorry. Assignment incorrectly written off.

See my answer here, Logic  :  https://web2.0calc.com/questions/can-someone-help-me-with-this_4

The arthmetic "formula"   for  the number of pages read on Day  N  is given by :

5  + (N-1)(.5)  =

5 + .5N - .5 =

4.5 + .5N

The first term  =  5

And the last term is    4.5 + .5N

So....we want to solve this :

(N/2)  [ first term + last term ] = 80

(N/2)[ 5 + 4.5 + .5N ]  = 80

(N/2) [ 9.5 + .5N ]  = 80       mutiply through by 2

N [ 9.5 + .5N ]  =  160

.5N^2 + 9.5N  - 160  = 0        multiply through by 2 again

N^2  + 19N - 320  = 0

The graph here (letting N = x)  shows two possible solutions  :  https://www.desmos.com/calculator/ze8gobvp4n

Taking the positive one...it will take about 10.75 days  =  11  days

https://www.desmos.com/calculator/1ilbxad7wu

Hannah starts at  (-4, 9)    walks 10 units to the left and then 7 units down

So.....she ends up at   ( -4 -10 , 9 - 7)   =  (-14, 2)

So....we need to find the slope  between    (-4,9)   and (  -14, 2)

We have

[ 2 - 9 ] / [ -14 - - 4]  =    [-7]/ [-10]  =7/10

Using either of the points to write an equation of the line we have that

y  =(7/10)(x - -4) + 9       simplify

y = (7/10)(x + 4) + 9      mutiply through by 10

10y = 7(x + 4) + 90

10y = 7x +28  + 90

10y  =  7x + 118 subtract 7x from both sides

-7x + 10y  = 118

If it has VARIOUS colors, it is due to diffraction of light from the surface   ....   kinda like a prism splits white light in to its component colors.

For a geometric series    S_n = a₁ (1-rⁿ⁾  / (1-r)

The common ratio is 1.5    (1.5 more pages each day)     a₁ = 5

80 = 5 (1- 1.5ⁿ⁾ / (1-1.5)

-40 = 5 (1-1.5ⁿ )

-8 = (1-1.5ⁿ)

-9 = -1.5ⁿ

9 = 1.5ⁿ       Take log of both sides

log9/log1.5 = n = 5.41 days ~~~ 6 days to read the 80 page book.

what gives light-reflecting objects various colors?  

You might think of it as which colors (which electromagnetic frequencies) the object absorbs.  Then, it reflects the rest. 

.

This is an arithemetic sequence. Use the arithmetic sequence formula to solve for N:

Solve for N:

N/2*[2*5 + 1/2*(N - 1) ]= 80, solve for N
1/2 N ((N - 1)/2 + 10) = 80

Multiply both sides by 2:
N ((N - 1)/2 + 10) = 160

Expand out terms of the left hand side:
N^2/2 + (19 N)/2 = 160

Multiply both sides by 2:
N^2 + 19 N = 320

Add 361/4 to both sides:
N^2 + 19 N + 361/4 = 1641/4

Write the left hand side as a square:
(N + 19/2)^2 = 1641/4

Take the square root of both sides:
N + 19/2 = sqrt(1641)/2 or N + 19/2 = -sqrt(1641)/2

Subtract 19/2 from both sides:
N = sqrt(1641)/2 - 19/2 or N + 19/2 = -sqrt(1641)/2

Subtract 19/2 from both sides:
 
N = sqrt(1641)/2 - 19/2 = 10.75, or about ~11 days to read the book 

It has to do with what frequencies of light the object reflects well, which has to do with the atomic structure of the object at it's surface.

It's really quite a complicated question as to why this surface reflects red well while this one reflects green well and to really get an answer

you're going to have to delve into quantum electrodynamics which is not a topic for the faint hearted.

For the first question, the probability of the first person selected being hopelessly romantic is 13/100, and the probability for the second person being hopelessly romantic is 13/100

Multiply these to find the probability of them both being hr: 13/100*13/100 to get 1.69% chance of them both being randomly selected.

(apply the multiplication of probabilities rule)

For the second one you should be able to add the probabilities of each person being hr (13/100+13/100), but the numbers wern't working for me. Maybe find find 1-(neither hr)-(both hr) to get the probability of only one of them.

CU, your approach to treat it like there are 100 people and 13 are hr dousn't work because there arn't only 100 people; there is a whole pupulation and your method undercounts because it has to deal with not being able to choose a person twice.

Let the center of the hyperbola be  at the origin  =  (0,0)

We will have this form

x^2           y^2

___    -    ____    =  1

a^2           b^2

c  = the distance from either of the focal points to the center   =  100  miles

And c^2  = 10000

For any point on the hyperbola, the differences of the distances from the two foci  to this point  = 2a

So

2a  = 180

a  =  90    ⇒   a^2  = 8100

And we have that

c^2  = a^2 + b^2

c^2 - a^2  = b^2

10000 - 8100  = b^2

1900  =  b^2

So  the equation becomes

x^2           y^2

____  -    _____   =      1

8100         1900

Here is a graph : https://www.desmos.com/calculator/5dyujkabt4

https://www.desmos.com/calculator/1ilbxad7wu

Thank you SO much!

2)

 12   2                 

 4    -2                 -24 - 8

______   =        _______  =     - 8

  1   2                   -2 +  6

 -3 - 2

1  12          

-3  4               4 + 36

_____  =       ______  =    10

1   2               -2 + 6

-3 -2

1)  We need to find  the inverse matrix  of

3  2

-1 5

This is given by

        1                5    -2      =        1        5   - 2    this goes in the first answer  box

__________      1     3              ___      1     3

3*5 - (2) (-1)                               17

  1               this goes in the second answer box

-23

   1              5   - 2            this goes in the third answer box

_____         1     3

 17

1            5     -2                1

___        1     3        x       -23      =

17

1         5(1) + (-2)(-23)

__       1(1)  + (3)(-23)

17

  1           51           this goes in the  fourth answer box

___        -68

 17

 3                     this goes in the last answer box

-4

BTW   " ....the coefficient of static friction is equal to the tangent of the angle at which the objects slide..."   

https://web2.0calc.com/questions/help_54963#r3

Yes slope = 7/10

Now you have two points (pick one)...     and sub into y = mx+b   ( m = slope)  to find the value of  'b'   and you will have your line eq

    you will need to re-arrange it to  ax + by = c   form......but I'm sure you can do it....  

https://web2.0calc.com/questions/help-please-thank-you_8#r1

The numbers in the equation look good to me. 

2y  =  f(2x) / 2  + 2       divide through by 2

y  = (1/4) f(2x) + 1

The point   (9/2, (1/4)(7) + 1)  =  (9/2, 7/4 + 1)  = (9/2, 11/4)  will be on the graph

The sum of the coordinates  is     9/2 + 11/4  =   18/4 + 11/4  =   29/4

We have these two equations

3  = (2)^2 + 2b + c

3 =  (4)^2 + 4b + c   simplify

3 = 4 + 2b + c

3  = 16 + 4b + c

-1  = 2b + c    ⇒   2 = -4b  - 2c    (1)

-13 = 4b + c      (2)

Add (1) and (2)  and we have

-11 = -c

c  = 11

\(v = \alpha\begin{pmatrix}2\\-3\end{pmatrix},~\alpha \in \mathbb{R}\\ 2\alpha - 3\alpha = 3\\ \alpha = -3\\ v = \begin{pmatrix}-6\\9\end{pmatrix}\)