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I'm sorry 'bout that. I know how that feels

Why? What's troubling you mate?

(-4-12.4)/-4 = -16.4/-4 = +4.1

what is the smallest power of ten that exceeeds 118,652.56484  

I'm not sure I understand the question.  Here's my answer to what I think it's asking. 

Start with the power of ten in the same order or magnitude. 

Consider 10⁵ = 100,000 ... that's smaller than 118,652.56484 so 10⁵ does not exceed it

The next power of ten is 10⁶ = 1,000,000 ... that's larger than 118,652.56484 so 10⁶ does exceed it.
 

The answer to the question is 10⁶   

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The total number of pennies will be: 1 + 2 + 4 + 8........and so on. Or: (2^23) - 1 =8,388,608 - 1 =8,388, 607 pennies.

8,388,607 / 100 pennies =$83,886.07 - Total amount of money after 23 nights.

Use the same reasoning in this one as in the previous question about the price of the car.

$9000 x [100% - 9%]^2 =$9000 x 0.91^2 =$7,452.90 - price of the car after 2 years. The decay is exponential because of 0.91^2.

On square 1 you have: 1

On square 2 you have: 2 or 2^(2 -1) =2 grains

On square 3 you have: 2^(3-1) =2^2 = 4 grains

On square 4 you have: 2^(4-1) =2^3 = 8 grains........and so on.

On square 28 you have: 2^(28-1) =2^27 =134,217,728 grains

Total numbers of grains on the board is just: (2^28) - 1 = 268,435,456 - 1 =268,435,455 grains.

The total weight of ALL grains of the board is:268,435,455 x 1 / 7,000 =38,347.92 or ~38,348 pounds

38,348 pounds / 2,000 pounds =~19.2 short tons!.

1 - The answer given to you: 479001600 comes from: [5 + 4 + 3] = 12! =479,001,600

2 - Also the answer given to you: 103680 comes from: 5! x 4! x 3! =17,280. Since the 3 categories can be arranged in 3! = 6 different way, then:17,280 x 6 =103,680 ways.

I have not answered the question that you have asked.  (I don't know how)

Like CalculatorUser I have just played with it.

Obviously the answer has to be between 0 and 0.5 but I know you want an exact answer with a prrof.

I graphed it using Geogebra.  On my computer the line CE can move to give different points of contact for E and I

To the accuracy Geogebra can give the maximum area is 0.25.

For 1:

\(a = -2\sin(\frac{-3\pi}{12})+1\)   Now a can be calculated in a straightforward way.

and   

\(1=-2\sin(3b)+1\)   Now rearrange to find b.

I'm sure you can find a and b from these (note that there will be an infinite number of values of b).

Use the same approach for questions 2 and 3.

"Suppose a single bacterium is placed in a bottle at​ 11:00 am. It grows and at​ 11:01 divides into two bacteria. These two bacteria each grow and at​ 11:02 divide into four​ bacteria, which grow and at​ 11:03 divide into eight​ bacteria, and so on.​ Now, suppose the bacteria continue to double every minute and the bottle is full at​ 12:00. How many bacteria are in the bottle at​ 11:39? What fraction of the bottle is full at that​ time?"

After 1 min there are 2¹ bacteria.

After 2 mins there are 2² bacteria.

After 3 mins there are 2³ bacteria.

...

After n mins there are 2ⁿ bacteria.

So after 39 mins there are 2³⁹ bacteria

After 60 mins there are 2⁶⁰ bacteria

Fraction full at 11.39 is thereferore 2³⁹/2⁶⁰ or  2^-21

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1.
The greatest integer function, \(\lfloor x\rfloor\)r, denotes the largest integer less than or equal to \(x\). For example, \(\lfloor3.5\rfloor=3\), and \(\lfloor -\pi\rfloor=-4\).

Find the sum of the three smallest positive solutions to \(x-\lfloor x\rfloor=\frac1{\lfloor x\rfloor}\).
Express your answer as a mixed number.

\(\text{Let $ \text{frac}(x) = x - \lfloor x\rfloor $ } \\ \text{Let $ x = \lfloor x\rfloor + \text{frac}(x) $ } \)

I assume:

\(\begin{array}{|rcll|} \hline x-\lfloor x\rfloor &=& \frac1{\lfloor x\rfloor} \quad | \quad x - \lfloor x\rfloor = \text{frac}(x) \\ \text{frac}(x) &=& \frac1{\lfloor x\rfloor} \\ \mathbf{\lfloor x\rfloor \cdot \text{frac}(x)} &=& \mathbf{1} \\ \hline \end{array} \)

\(\begin{array}{|rclcl|c|} \hline \mathbf{\lfloor x\rfloor } &\cdot& \mathbf{\text{frac}(x)} &=& \mathbf{1}& x = \lfloor x\rfloor + \text{frac}(x) \\ \hline 2 &\cdot& \dfrac{1}{2} &=& 1 & 2\dfrac{1}{2} \\ \hline 3 &\cdot& \dfrac{1}{3} &=& 1 & 3\dfrac{1}{3} \\ \hline 4 &\cdot& \dfrac{1}{4} &=& 1 & 4\dfrac{1}{4} \\ \hline 5 &\cdot& \dfrac{1}{5} &=& 1 & 5\dfrac{1}{5} \\ \hline 6 &\cdot& \dfrac{1}{6} &=& 1 & 6\dfrac{1}{6} \\ \hline &\ldots &&&&\ldots \\ \hline \end{array} \)

The sum of the three smallest positive solutions is \(2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4} = 10\dfrac{1}{12} = 10.08\overline{3}\)

\(4\sqrt{x^2+y^2+z^2} = 28\\ x^2 + y^2 + z^2 = 49\\ \text{The other two equations are correct}\)

\(z^2 = 49-40 = 9\\ z=3\\ y^2 = 45-9=36\\ y=6\\ x^2 = 49-36-9 = 4\\ x=2\\ xyz = 36\)

Thank you, CPhill !

\(\text{$15=3\cdot 5$ so $15n$ being a perfect square means $n=15 k^2$}\\ \text{We want all of these up to 1000}\\ 15k^2 \leq 1000\\ k^2 \leq 66\frac 2 3\\ k\leq 8\\ \text{So we have $n = 15,60, 135, 240,375,540,735,960$}\)

\(\text{I assume the books of a given type are indistinguishable}\\ \text{You have 12 books. If they were all distinguishable there would be $12!$ arrangements}\\ \text{As they are not we have to reduce this number by the number of permutations of each type}\\ N = \dfrac{12!}{5!\cdot 4!\cdot 3!} = 27720\)

xy + xz + yz   = 0         (1)

yz = -(xy + xz)   ⇒    -yz  = +xy + xz        (2)

xz = - (xy + yz)   ⇒   -xz =  + xy + yz        (3)

xy = - (xz + yz)    ⇒   -xy =  + xz + yz       (4)

 1                          1                              1

______ +         ________      +         ________    =   

x^2-yz              y^2 - xz                     z^2 - xy

 [ sub in (2), (3) and (4)  in the denominators ]

         1                         1                                   1

___________  +  _____________  +     _____________   =

x^2 + xy + xz       y^2  + xy + yz                  z^2  + xz + yz

   1                              1                               1

__________   +      ___________   +   _____________   =

x ( x + y + z)            y ( x + y + z)              z(x + y + z)

       yz                                   xz                                xy

____________     +          ____________   +     ____________      =

xyz(x + y + z)                    xyz(x + y + z)              xyz ( x+ y + z)

( xy   +  xz   + yz  )

________________    =         

   xyz ( x + y + z)

sub in (1)  for the numerator

      (   0   )

______________     =           0

  xyz ( x + y + z)

x^(3/2)  [ x^(1/2)   - x^(-1/2) ]        [distribute  x^(3/2)  over both terms in the brackets ]

x^(3/2) * x^(1/2)   -  x^(3/2)* x^(-1/2)        [ use the rule that  x^m * x^n  = x^(m + n) ]

x^( 3/2 + 1/2)  -  x^(3/2 - 1/2)       [simplify the exponents ]

x^2   -  x

Notice Cphill's answers

they are ODD perfect squares except for 4, and ONLY prefect squares.

Answers like 81 won't work because it is a perfect power of 4 and 64 is a perfect cube.

That's the only ones I could think of  !!!!

Thanks c-phil!

How many numbers n less than 100 have exactly one divisor other than 1 and n?

n     divsors

4       1   2   4

9       1   3   9

25     1   5  25

49     1   7  49