CPhill

log₉(x^2+6x)=log₉ (16)

Since the bases are the same, we can solve for......

x^2 + 6x = 16      subtract 16 from both sides

x^2 + 6x - 16 = 0     factor

(x - 2)(x + 8) = 0        and settting each factor to 0 . we have that x = 2 or x = -8

Both of the solutions are good, since neither  produce a negative log

1 quart = 1/4 gallon

1/3 of this is antifreeze  = (1/3)(1/4) =  1/12 gallon

And    2/3 is water......so (2/3)(1/4) = 1/6 gallon

And the half-gallon  contains   3/4 antifreeze = (1/2)((3/4) = 3/8 gallon

And the half-gallon  contains (1/4) water =   (1/2)(1/4) = 1/8 gallon

So...the  ratio of antifreeze to water is  [  1/12 + 3/8] / [1/6 + 1/8]  = [11/24] / [14/48] = [22 / 14 ] = [11/ 7] or 11 : 7

Thanks, Melody.....

We have

x^3 + (x + 1)^3 + (x + 2)^3  = (x + 3)^3  ...  and expanding, we have

x^3 + x^3 + 3 x^2 + 3x + 1 + x^3 + 6x^2 + 12x + 8  = x^3 + 9x^2 + 27x + 27  ....  simplify

3x^3 + 9x^2 + 15x + 9  = x^3 + 9x^2 + 27x + 27

2x^3 - 12x - 18 = 0

x^3 - 6x - 9 = 0   Using the Rational Zeroes Theorem, the possible roots of this are ±1,  ±3  and  ±9 

And  x = 3 is one solution

Using synthetic division. we have

3        1    0   - 6    -9

                3     9     9

          1    3    3      0

And the remaining polynomial  is   x^2 + 3x + 3   which has no "real" roots

So the integers are   3,4, 5    and 6

Check.....    3^3 + 4^3 + 5^3   =  27 + 64 + 125 = 216  = 6^3

One hour...... To see why......Let's suppose that you either awoke at 12:30AM  or at 1AM......At each of these times you would have heard a "one."

If you awoke at 12:30AM.....the next "one" you would hear would be at 1AM......But you couldn't be sure if this "one" was to signal 1AM or 1:30AM......But, upon hearing the third "one"- 30 minutes later -  you would know that this sequence could only happen at 12:30 AM, 1AM and at 1:30AM

Similarly...if you awoke at 1AM......the next "one" you would hear would occur 30 minutes later - at 1:30AM. But, as before.......you couldn't be sure if this "one" was to signal 1AM or 1:30AM.......so you would have to wait until 2AM - when the clock struck "two" - to ensure the correct time.

In either case, you would have to stay awake one whole hour.......

This polygon is comprised of 30 congruent triangles, each with a central angle of 12°  

And using trig, each has an area of......

(1/2)(20)^2 sin(12°) = 41.5823381636 sq ft

So, 30 times this = 1247.470144908 sq ft  = 1247.5 sq ft  (rounded to the nearest tenth)

Scary......!!!!

Note....

5 log 2  =   log 2^5   = log 32  .... so  ...... n = 5