CPhill

If group  A  can assemble 16 cars in 8 hours, they can build 2 cars per hour

So  each hour in this group....we have 5 labor hours for the men and 6 labor hours for the women

And  group B can assemble  38 cars in 15 hours, so they can build 38/15 cars per hour

And each hour in this group, we have 6 labor hours for the men and 8 labor hours for the women

And  call the portion  of a car built  by a man  in one hour 1/x

And call the  portion of a car built by a women in one hour 1/y

So....we have this system

5 (1/x)  +  6 (1/y)   = 2

6(1/x) + 8(1/y)  = 38/15

Solving this system produces  x  = 5   and y  = 6

So....a man can produce 1/5 of a car per hour and a women produce 1/6 car per hour

So  for group C which consists of 8 men and  3 women.....

The number of cars produced in 10 hours is : 

8 man hrs per hour * (portion of a car produced by a man in one hour) * 10 hours   plus

3 women hrs per hour * (portion of a car produced by a women in one hour) * 10 hours  =

8 (1/5) (10)  + 3(1/6)(10)  =

80/5  +  30/6  =

16  +  5  =

21  cars

(x + 7 ) / ( x^2 - x - 2)   =   A  / (x - 2)   +  B / ( x + 1)

We can use partial fractions, here

We can write

( x + 7)  /[( ( x + 1)( x - 2)]  =  A / ( x - 2)  + B / ( x + 1)

Multiply through by  (x + 1) ( x -2)

x + 7  = A ( x + 1)  +  B( x - 2)    simplify

x + 7   = Ax   +  A   +   Bx - 2B

x + 7  =  ( A + B) x  +  ( A - 2B)

Equating coefficients, we have

A + B   = 1

A - 2B  = 7      subtract the first equation from the second 

-3B =  6

B  = -2

And  we can find  A  as  A + B  = 1 ⇒    A - 2  = 1   ⇒  A  = 3

So  (A, B)   = ( 3, - 2)

Let's take this by stages

r^3m                        r ^4m                                                        r  ^4m/n  

______   =   _______        raising all this to 1/n  =      ______

r^-m * t⁴ⁿ           t ⁴ⁿ                                                                       t  ⁴

And

r  ^1/n                                                  (   t ^2/m )^m                  t ²

_____  raised to the  -m    =    _______   =        ______ 

  t  ^2/m                                               (    r ^1/n ) ^m               r ^m/n

So....putting all this together, we have

r ^4m/n   *  t²                 r ^3m/n

_________     =    ________

t⁴  *   r ^m/n                    t ²

The divisors of 40 are

1  2  4  5   8  10   20   and  40

But  x  is a positive integer..  and since x + 3  divides 40......the possible values for x are

1   2   5    7   17    37

So  when

x = 1       LCM  =  4  ( impossible....the LCM is at least 40)

x = 2       LCM  =  10

x = 5       LCM   = 40

So  x  =  5   and  the  other  integer  is  8

OK....glad to help   (if I can )  !!!

We  need to take the  cube root of the  ratio  of  the  volume of the larger solid to the smaller to  find the scale factor

So  we have   ∛ [ 1125 / 576  ]  = ∛  [ 125/64 ] =   5/4

So.....the correseponding side of the larger solid  =

Side length of the smaller solid  *  ( scale factor of larger solid to the smaller )

12  *  (5/4)  =

60  / 4  =

15  inches

Asd an adddendum.....when we are talking about volume comparisons of similar solids....we are either taking a cube root to find a scale factor....or we are cubing the scale factor to find a voliume

For surface area......we   are either taking a square root to find a scale factor, or we are squaring the scale factor to find a surface area

The surface area  of  Y  to X  is   (  891/ 539)  = 81/49

For any dimension from Y to X.....we can take the square root of 81/49  = 9/7

So....this is the scale factor  for  each dimension from Y to X

So....the height of Y  will be  =

Height of X  * ( scale factor of Y to X )   =

28  * (9/7) =

28 / 7  *  9   =

4  * 9  =

36 cm

The  scale factor  of the larger pyramid to the smaller is 12/9  =  4/3

The volume of the larger will be equal to

Volume of the smaller pyramid  *  ( scale factor ) ^3

So we have

162  * (4/ 3)^3  =

162  *  (64  / 27)  =

384  in^3

The scale factor from B  to A  is  (12/18)  = (2/3)

So...the surface area  of  B  =   the surface area  of A  *  the scale factor of B to A squared

So we have

1836  *  (2/3)^2  =

1836 *  (4/9)  =

816  cm^2