CPhill

26.16 - 7.647  = 18.513

Thanks,  Landry.......here's another formula....if we know the side length

We have 5 equal isoceles triangles.......each with an apex angle of 72°

If we know the side length, the height of each triangle can be found, thusly:

tan (36) = (1/2)s / h

h = s / [2 tan 36]  =  (1/2)s*cot36

So....the area of one triangle =

(1/2)s * (1/2)s * cot 36  =

(1/4)s^2 * cot(36)

So....the total area =

(5/4)s^2*cot(36)  =

(5/4)s^2 * tan(54)

I'm  assuming this is supposed to be :

135+3x=T

 25+5x=T

Since each equation equals T, we can just set them equal.....and we have

135 + 3x   = 25 + 5x    subtract 3x, 25 from both sides

110  = 2x     divide both sides by 2

55 = x

Actually.....since were solving an equation, we need to take both roots....so  ...

t = ± 11    is correct

Here's the answer....although someone else can probably present it in a more straightforward way........!!!!

Seat Thomas (T) in chair 1  and Lily (L) in any chair from 3 - 7....she has 5 choices

Richard/Hang (RH) can be considered as one entity and can be arranged in 2 ways each

And the other 4 people can be arranged in 4! =  24 ways.....so we have

T 2 L 4 5 6 7 8 ... RH (4 choices  x 2 arrangements each )  x  24  = 192 x 5 choices for Lily = 960 

 If  Thomas is seated in chair 1 and Lily in chair 8 , we have

T 2 3 4 5 6 7 L ...RH (5 choices x 2 arrangements) x 24  = 240

So....when Thomas is seated in chair 1 there are

960 + 240    = 1200  arrangements possible

Next ....seat Thomas in chair 2  and Lily in any of the chairs 4 - 7....she has 4 choices

Following the above notations, we have

1 T 3 4 5 L 7 8    ...RH (3 choices  x 2 arrangements) x  24  = 144 x 4 choices for Lily  = 576

And when Thomas is seated in chair 2 and Lily in chair 8 we have

1 T 3 4 5 6 7 L  ...  RH 4 choices x 2 arrangements X 24 = 192

So....when Thomas is seated in chair 2 there are

576 + 192  = 768  arrangements possible

When Thomas is seated in chair 3, we have the following :

L 2 T 4 5 6 7 8 ...  RH  (4 choices  x  2 arrangements)  x 24   = 192

1 2 T 4 [Lily can occupy any chair 5 - 7]  8   ...  RH (3 choices x 2 arrangements) x 24 x 3 choices for  Lily = 144 x 3 

1 2 T 4 5 6 7 L ....   RH (4 x 2) x 24 =  192

So when Thomas is seated in the third chair we have

192*2 + 144*3  = 816  arrangements possible

And this number of arrangements wll be true when Thomas occupies any of the chairs 3 - 6

So....we have 816 (4) = 3264 possible arrangements when Thomas occupies chairs 3 - 6

When Thomas occupies chair 7, there will be the same number of arrangements as when he occupies chair 2  = 768

And when he occupies chair 8, there will be the same number of arrangements as when he occupies the first chair  = 1200

So we have

1200(2) + 768(2) + 816(4)  = 7200 total arrangements

I assume you want to express

7 x 7 x 5 x 5 x 5 x 5   with exponents  ???....if so....

Write   7 ..   and we have two of them  = 7²

Write 5 ....  and we have  four of them  = 5⁴  

So....the answer  [ without final evaluation] is :

7² * 5⁴    

In general   .....  "a"  multiplied "b"  times    ...... is just .....  a^b

If C'x = -.05 + .001x.......then  the variable cost part of the function becomes :

-.05x + .0005x^2

And the fixed cost part of the function is just  5000

So.....the function is :

C(x)  = -.05x + .0005x^2 + 5000

Proof :

Note that the cost to produce 500 lbs is

C(500)  = $5100

And the cost to produce 5000 lbs is

C(5000) = $17250

So....the extra cost to increase production from 500 to 5000 lbs = $[17250 - 5100] = $12150

Area = Length x Width  .......so

2/3 =  Length x 1/2    multiply both sides by 2/1

(2/1)(2/3)  = Length

4/3 km   = Length

No prob......