CPhill

sqrt (108) = sqrt (36 * 3)  =  6sqrt(3)

sqrt (12)  = sqrt (4 * 3)  = 2sqrt(3)

So

sqrt (108)  - sqrt (12)  =

6sqrt (3)  - 2sqrt(3)  =

4sqrt(3)  =

sqrt (16 * 3)  =

sqrt (48)    ......so    N  = 48

We have a point (4,8) ......  and the slope of the curve at that point  = 6

So...the equation of the tangent line to that point  will have the same slope as the curve at that point....and we have

y - 8  = 6(x - 4)

y - 8  = 6x - 24      add 8 to both sides

y  = 6x - 16

Check out Graham's number.......

See my answers, here :    http://web2.0calc.com/questions/a-large-lot-of-items-includes-10-defectives-thus-the-chance-of-selecting-a-defective-item-is-0-1-and-conversely-the-chance-of-selecting-a-good-item#r1

Probability of selecting 1 defective item from any 10  =

C(10,1) * (.10) * (.90)^9  = about .3874

Probability of selecting 2 defective items from any 10  =

C(10,2) * (.10)^2 * (.90)^8  = about .1937

So....the probability of getting 1 OR 2 defectives  =  about  .3874 + .1937  = .5811

The slope at any point is given by the derivative of the function  =

8 - 15e^x

So at x = 0  , the slope is   8 - 15e^0   =  -7  = m         and  b = -14

And the equation of the tangent line  is

y = -7x -  14

Here's the graph with the tangent line  ...  https://www.desmos.com/calculator/zw5yhfi2ar

Melody copied the problem  wrong......I WIN   !!!!!!!!

Mine is correct guest.....

But mine was MORE competitive, DS.......!!!!!!!

Rats....!!!....Melody beat me to it........!!!!!!!

[ She probably cheated......LOL!!!!!]

2(x-3)^2 -8=0   divide through by 2

(x - 3)^2  - 4  = 0       factor as   a difference of squares

[ (x - 3) + 2]  [ (x - 3)  - 2]  = 0

[ x - 1]  [ x - 5]  = 0      and setting each factor to 0 , x =  1  or x = 5