CPhill

Thanks, CentsLord and LuckyDucky   !!!

Here's my approach  :

Bob has finished   (8/10) = 4/5 of the race  and Carol has finished (6/10) =3/5  of the race when Anna finishes

So....in 4/5  of the race  he has as 200 meter lead on Carol

So...in each (1/5) of the race  he gains 200 /4  = 50 meters on Carol

So....for the last 1/5  of the race he picks up another 50m on Carol

So....when he crosses the finish line, he is  200 + 50  =  250 m  ahead of Carol

∫ 8 e^{x+ 1}  dx  =

8 ∫ e^{x + 1}  dx  =

8 *  ( derivative  of x + 1) * e^x+1  +  C   =

8 * ( 1) * e^x+1  +  C  =

8e^x+1  + C

The  best  way to see this is just  to realize the anti-derivative properties of  both functions

e^ax  has  the anti-derivative  of   (1/a) e^ax

So......letting  a  = 3  we have that

e^3x  =  (1/3)e^3x  

Similarly...lettig a  = -1   we have

5 (e ^-x)  =  5 ( e ^-1x)  =  5 ( 1/-1) (e^-1x)  =  5 (-1)e^-x  =  -5e^-x

ln 3

∫      e^x  dx    =     e ^{ln 3}   -  e ^{ln 2}

ln 2

Note that  e^{ln 3}  =  3     and   e ^ln2   =  2

So...the area  is just  3  - 2   =    1

∫  e^3x + 5e^-x  dx   =

(1/3)e^3x  +   (-5)e^-x    +  C

√2  * √6    * √110  * √165

√2 * √2 * √3 * √11 * √10   * √33 * √5   = 

√2 * √2 * √3 * √11 * √2 * √5  * √3 * √11 * √5   =

√2 * √2 * √2  * √3 * √3  * √5 * √5 * √11 * √11  =

2 * √2 * 3 * 5 * 11  =

330√2

Just a slight mistake, LuckyDucky

25 percent  "of what"   = 19

"of what "  means "multiply"

So...let "of what"  = x       and we have that

.25 x  =19       divide both sides by .25

x  =76

Welcome....!!!

Take 1/2  of 22  =  11

Square this  = 11^2  =  121

That's what goes into the box

Proof

(x + 11)^2  =

(x + 11)  ( x + 11)  =

x^2  + 22x + 121

OK  we have this

6 + b + c   = 18

b + c  =  18 - 6

b + c  = 12       (1)

Also

6^2 + b^2  = c^2      [ Pythagorean Theorem ]

36 + b^2 = c^2

36  = c^2 - b^2

36 = ( c - b) ( c + b)         sub in (1) for b + c

36 = (c - b) ( 12)    divide both sides by 12

3  = c - b    rearrange as

b = c - 3     sub this into (1)  for b

c - 3 + c  = 12

2c = 15

c =7.5