geno3141

Let  x  =  number of lessons

Members:  120 + 20x

Non-members:  35x

For them to be equal:     120 + 20x  =  35x

                                                 120  =  15x                      [subtract 20x from both sides]

                                                     8  =  x                          [divide both sides by 15]

If you are just supposed to simplify it:

We know that:  log₂2  =1 , so  log₂4  =  log₂2²  =  2·log₂2  =  2·1  =  2

Therefore:        2·log₄7  =  2·(log₂7 / log₂4)  =  2·(log₂7 / 2)  =  log₂7

Also:                log₂98  =  log₂(7²·2)  =  log₂7² + log₂2  =  2·log₂7  + 1

Putting these together:  (log₂98) - (2·log₄7) - 2  =  (2·log₂7  + 1) - (log₂7) - (2)

                                                                            =  log₂7 - 1

                                                              or           =  log₂7 - log₂2  =  log₂(3.5)

                                                               or:         =  log₄49 - 1  =   log₄(12.25)

Although EP is correct, if the question were:  How many different hands can be constructed that contains 2 kings and 3 aces?

then you would need to select 2 kings from 4 kings and 3 aces from 4 aces:  ₄C₂ x ₄C₃  =  6 x 4  =  24

I believe that the flaw occurs because you handle "S" as if it were a real number.

There is no real number (or whole) number that represents the sum of 1 + 2 + 3 + ...

I wrote a little program on my TI-84 that (I hope) has gives the number of possibilities..

Using the digits "1", "2", and "3" instead of colors, the results are:

12123     21213     31212

12132     21231     31231

12312     21313     31232

12313     21321     31312

12323     21323     31321

13123     23121     32121

13132     23123     32131

13212     23131     32132

13213     23213     32312

13232     23231     32321

I haven't come up with a neat explanation, though.

This could be done in two ways:     orange - purple - orange     or     purple - orange - purple

orange - purple - orange  =  (4/9) x (5/9) x (3/9)  =  60/729

purple - orange - purple   =  (5/9) x (4/9) x (4/9)  =  80/729

To find the final probability, add these two probabilities together.

Use the Pyhagorean Theorem from the origin to the top of each tree.

In one case, distance = sqrt( 10² + 14² ).

In the other case, distance = squrt( 13² + 9² ).

Find each distance to determine the answer to the problem.

Since  WY = WY  and  WZ = WX  and angle(XWY) > angle(ZWY),

                                 the side opposite angle(XWY) > the side opposie angle(ZWY).

log(24, 6)  =  log₆24  =  log₆(6 · 4)  =  log₆6 + log₆4  =  1 + log₆4   

log(24, 4)  =  log₄24  =  log₄(4 · 6)  =  log₄4 + log₄6  =  1 + log₄6

[1 - log(24,6) ] · [1 - log(24, 4) ]  =  [ 1 - (1 + log₆4) ] · [ 1 - (1 + log₄6) ]

                                                   =  [ - log₆4 ] · [ - log₄6 ] 

                                                   =  log₆4 · log₄6

Changing these both into base 10:  =  [ log4 / log6 ] · [ log6 / log4 ]  =  1

The water cooler:  V  =  pi · radius² · height  =  pi · (9)² · (22)  =  1782 pi.

The paper cup:     V  =  (1/3) · pi · radius² · height  =  (1/3) · pi · (2)² · (3)  =  4 pi.

Can you finish it from here?  [If the answer does not come out to be an exact number, round down.]