geno3141

There is a neat website that factors numbers for you:

I fully agree with the 1/3, 2/3 probabilities.

At the beginning, the player has a 1/3 chance of getting the correct door; the host has a 2/3 chance of having the correct door.

The host gives the player the option of trading sides. If the player trades, then the player gets the host's 2/3 chance of winning. (Showing an incorrect door doesn't change the chances; the host must always start with (at least) one losing door. All this does is fake people into believing that the chances are 50-50.)

Switching from the one that you originally chose to the one not shown by the host gives you the best overall probability of winning.

The factors of 60 are:  1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Since only 50 books can be on a shelf, the usable factors are:  1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30

Using 6 shelves:  6 x 10 = 60

Using 5 shelves:  5 x 12 = 60

Using 4 shelves:  4 x 15 = 60

Using 3 shelves:  3 x 20  = 60

Using 2 shelves:  2 x 30 = 60

If there is no compounding, it will take 20 years.

The simple interest formula:  Interest  =  Principal x rate x time

To double, the amount of interest must equal the amount of principal, so, using the formula:

                                           4000  =  4000 x 0.05 x time

                                           4000  =  200 x time

                                              20  =  time

However, not having compounding is unusual; if compounded:

The compound interest formula:  A = P(1 + r/n)^(n·t)

A = final amount = 8000        P = principal = 4000         r = rate (as a decimal) = 0.05

n = number of times compounded per year         t = time (years)

       8000  =  4000(1 + 0.05/1)^(1·t)

       8000  =  4000(1 + 0.05)^t

            2  =  (1.05)^t

      log(2)  =  t·log(1.05

             t  =  log(2) / log(1.05)  =  14.2 years

2³ = 8         4² = 16          2² = 4

(2³ + 4²) / (2² x 3)    =    (8 + 16) / (4 x 3)  =  24 / 12  =  2

f(x)  =  2x³ - 3x² - 18x + 8

f(1)  =  2(1)³ - 3(1)² - 18(1) + 8

       =  2 - 3 - 18 + 8

       =  -11

90  =  2 x 45

     =  2 x 3 x 15

     =  2 x 3 x 3 x 5         or      2 x 3² x 5

As decimals: 1 1/4 = 1.25  and 7/8 = 0.875

4(1.25) + 2(0.875)  =  5.00 + 1.75  =  6.75 feet long

So, the board must be at least 6.75 feet long.

g(t)  =  -(8/27)t³

Replace 'g(t)' with 't', and 't' with 'g^-1(t)':

--->   t  =  -(8/27)[g^-1(t)]³

Now, solve for g^-1(t):

  --  Divide both sides by -(8/27)         --->   -(27/8)t  =  [g^-1(t)]³

  --  Find the cube root of both sides    --->  -3/2·t^1/3  =  g^-1(t)

--->  g^-1(t)  =  -3/2·t^1/3