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No table appears.......  

See the pic of the situation, here :

Note that in triangle ABF.....that  AF  is a base  and DG is  a segment parallel to this base....but a segment in a triangle that is parallel to a base splits the sides of the triangle proportionally

Therefore  :   BD / DA  =  BG/ GF

But D  is the midpoint of AB...so.... BD  = DA

But....this also means that  BG  =  GF

And  DGC  forms another triangle with DG  as a base  and EF a segment parallel to this base

And for the same reason as above :

DE / EC  =  GF / FC

But  E  is the midpoint of  CD....thus....DE  = EC

But...this also means that GF =  FC

But....it was proved that  BG =  GF

So.....by transitivity ...... FC  also  equals BG

Therefore    BG  = GF  =  FC

1 / √2    +  3 / √8   +  6 / √32   

1 / √2  +  3 / [ 2 √2]  +  6 / [ 4 √2 ]

1 / √2    +  3 / [ 2 √2]  + 3 / [ 2 √2 ]

2 / [ 2 √2]  +  6 / [ 2 √2]  

8 /  [ 2 √2 ]  

4  / √2  

4 *√2  /  [ √ 2 * √ 2 ]

4 * √2  /  2  

(4/2) √2

2√2

( 3x^2  - 2x  + 2  ) -  ( x^2 + 5x - 5 )  =   a negative outside a parenthese changes the signs on the terms inside

3x^2  - 2x   +  2   -  x^2  - 5x  +  5  =  combine like terms

3x^2 - x^2 - 2x - 5x  + 2 + 5  = 

2x^2  - 7x  + 7

Nice Job, Well Done 

( 4x  - 3  + 2x^2 )  +   (2x + 1)  =

4x + 2x  +  2x^2  -  3   +  1  =

2x^2 +  6x  -  2

 −6x³y² + 5x⁴ − 7z

The degree  will be the greatest sum of the exponents on any term

So....    3  + 2    =    degree 5

30 = 10x + 6y - 4x      simplify

30  = 6x + 6y       divide through by 6

5  = x  +  y

So....  b and d   are the answers  where the sum of the coordinates  = 5

Quartic =  highest power on x  = 4

"Tri" nomial  =  3 terms

So

Quartic Trinomial

All are polynomials but the third one

Girls  =  (3)  / (4 + 3)  * 28  =    (3/7) * 28  =  3 * (28/7)  =  3 * 4  =  12

The width of a rectangle is 4 cm less than the length. If each dimension were increased by 3 cm, the width of the new rectangle formed would be two thirds the length of the new rectangle. Find the dimensions of the original rectangle. 

Let the oiginal length  = L

Let the original width = W =  L  - 4

When  each dimension is increased by 3  we have the following dimensions :

New W = (L - 4) +  3    =  L -  1    and   New L  =     L + 3

So...the new width is (2/3)  the new length

So

L -  1  =  (2/3) (L + 3)       simplify

L - 1    (2/3)L + 2        add 1 to both sides, subtract (2/3)L from both sides

(1/3)L  =  3     multiply both sides by 3

L  =  9 cm  =  original length

L - 4  =  W =  9 - 4   =  5 cm  = original width

The formula  is

y  = 477 (1/2)^t/37   

y  = 477(1/2)^6/37  ≈   426.288 kg     [  second answer  ]

6²ⁿ  =  800      take the log of both sides

log 6²ⁿ   = log 800      and by a log property, we can write

2n * log 6  = log 800

n *  ( 2 log 6)  =  log 800           divide both sides by 2 log 6

n  =  log 800  /  [ 2 log 6 ]   ≈  1.87

ln 5  +  ln  (2x)  =  5        we can write

ln ( 5 * 2x)  =  5

ln (10x)  =  5       in exponential form ,we have

e^5  = 10x         divide both sides by 10

e^5 / 10  =  x  ≈  14.84

I'll check it with a higher power.

1+1= = 2       Yes.!!!!

...I have consulted with THE supreme being (web2.0calc). ......IT says your answer can now be validated with a tick  !!  

Thank you so much this was super helpful!

Part b

Using  the result from the first part, we have that

(n) ( 2a  +  n - 1)  /  2  =  100     multiply both sides by 2

(n) (2a + n - 1)  =  200

Factors  of 200  are      1 | 2 | 4 | 5 | 8 | 10 | 20 | 25 | 40 | 50 | 100 | 200

So we have the following possibilities for

n               (2a +  n  - 1)     =       200

2                  100                   a  is not an integer  if n  = 2

4                    50                   a is not an integer if n  = 4

5                    40                   a  =  18

8                    25                   a  =  9

10                  20                   a is not an integer if n  = 10

We can stop here....all other possible factor combinations for 200 result  in negative values for a

So....  ( a , n)    =   ( 18, 5)   and ( 9 , 8)     

Part (a): Find the sum

               a + (a + 1) + (a + 2) + ... + (a + n - 1)

in terms of a and n. 

We can write

( n ) a   +   ( 1 + 2 + 3 + ... +  (n - 1) )

The sum of the second part is

(n - 1) ( n)  / 2

Putting these together, we have that the sum  is :

(n)a  +  (n) ( n-1) / 2    =

(n)   [   a  +  (n - 1) / 2  ]  =

(n) ( 2a  +  n - 1)  /  2

 y^4 - 3y^2 - 270  =  0       factor

( y^2  - 18) (y^2  + 15)  =  0

Only the first factor  provides the real solution  of   y  =  √18 cm  =  height of the triangle

And...since the altitude of this triangle  sets up a 30-60-90 triangle, the side of the equilateral triangle  =  2√18 / √3  =   2√6  =  √24 cm

So....the area  of this  triangle  is  =

(1/2)base * height  =

(1/2)√24 * √18

(1/2) 2√6 * 3√2  =

3√12   =

6 √3  cm ^ 2

Here's one method  [ but maybe not the fastest or easiest ]

Let  AC   =  √ [ 24^2  + 18^2]  =  √ 900  =  30

So....let  A  = (-15, 0)  and C  =  (15, 0)

And we can find B and D  by intersecting the circles

(x + 15)^2 +  y^2  = 24^2      and

(x - 15)^2  + y^2  = 18^2

So....the intersection of  these (arbitrarily)  gives  B = (4.2, 14.4)  and D  = (4.2, -14.4) 

Here's a pic :

Let the center of the circle   lie at   ( E, 0)  [ I'm using E for the center instead of O ]

The slope of AB  is  [ 14.4] / [ 4.2 - - 15]  =  14.4  / 19.2  =  3/4

And the equation of this line segment is 

y =  (3/4)(x -  -15)

y =  (3/4)x + 45/4

4y = 3x + 45

3x - 4y + 45  =  0

And the slope  of BC  is  -4/3

And the equation of this line segment  is

y  =  (-4/3)(x - 15)

y =  (-4/3)x + 20

3y  = -4x + 60

4x + 3y - 60  =  0

And the distance   between  a point (m, n)  and the line  Ax + By + C = 0  is given by

l  Am + Bn  + C l   /  √ [ A^2 + B^2  ]

So  we want to  equate  these and solve for  E

l  3E  - 4 (0)  +  45 l  / √ [ 3^2 + (-4)^2]   =  l 4E + 3(0)  - 60 l / √[4^2 + 3^2 ]

l  3E +  45 l  /  5   =  l 4E - 60 l / 5

We have these two possible equations

3E + 45   =  4E - 60

E  =   105        reject

Or

3E + 45  =  -  [ 4E - 60 ]

3E + 45  =  -4E + 60

7E  =  15

E =  15/7

So....the center  of  of the circle  is  (E, 0)  =  (15/7, 0)

And using   l 3C + 45 l  /  5....the radius of the circle is

l 3 (15/7) + 45 l  /  5  =    l 45/7  +  45 l  / 5  =   45/35 + 9  =  9/7 + 9  =  72/7 cm

So.....the equation of the inscribed circle is

(x - 15/7)^2  + y^2  =  (72/7)^2

num of people after   0   years   =   2000

num of people after  10  years   =   2000(1 + 0.014)¹⁰   =   2000(1.014)¹⁰

The number you calculated is the population after 10 years...now we need the avg rate of change.

avg rate of change   =   \(\frac{\text{change in num of people}}{\text{change in num of years}}\)

avg rate of change   =   \(\frac{2000(1.014^{10})\,-\,2000}{10\,-\,0}\)     people/year

avg rate of change   ≈   29.831   people/year     ≈     30 people/year   

Well....I'm pretty sure it's right, but....my answer still hasn't been validated with a checkmark!

Here is a graph of log x

1   true

2   true

3   false

4   true  

Amount of solution A  =   x

                               B =   10 - x

x (.2) + (10-x)(.04) = 10 (.08)

.2x + .4 -.04x = .8

.16x = .4

x = 2.5 oz = amount of A     (10-2.5) = B = 7.5 oz 

Note

3¹  = 3

3²  =  9

3³  =  27

So.....

3ⁿ  appears to model the distance of the fall after n seconds   {Answer A  ]