CPhill

x^2 + x - 10 = 0   this won't factor....using the onsite solver, we have

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{10}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{41}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{41}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{3.701\: \!562\: \!118\: \!716\: \!424\: \!3}}\\
{\mathtt{x}} = {\mathtt{2.701\: \!562\: \!118\: \!716\: \!424\: \!3}}\\
\end{array} \right\}$$

It's just C(9,5) = 9! / [(9-5)! * 5! ] = 9! /[4! * 5!] = [9 x 8 x 7 x 6] / [ 4 x 3 x 2 x 1] =

[3 x 2 x 3 x 7] = 126 ways

We are trying to solve

27*∑ (1/3)^(n-1)   from 1 to n =  364/9

WolframAlpha shows that 6 terms are needed

Here are the sums from n = 1 to n = 6

27 + 9 + 3 + 1 + 1/3 + 1/9 = 364/9

Your answer is just

4 x 17 x 30.......????

Ah....I see...sorry!!

Yep...nice one, Sasini....!!!

Thanks, Melody......to be honest...I don't even know if this is exactly the correct answer, or not...I just kind of "winged it."

2^x = 225    take the log of both sides

log 2^x = log 225   and we can write

xlog 2 = log 225       divide both sides by log 2

x = log225/log2 = 7.8137811912170377

And yet, here again is another possibility....if you have had calculus, we know that the derivative = 0 where the turning points occur,

Therefore, let the derivative of the function be (x -3) (x +2) = x^2 - x - 6.

Then, again.... this may only make sense if you have had calculus.......our basic function would be:

F(x) = (1/3)x^3 - (1/2)x^2 - 6x

This function will have a y intercept of 0, so we  just need to append a "10" at the end to give us a y intercept of 10.

So....our "translated" function would be...

F(x) = (1/3)x^3 - (1/2)x^2 - 6x + 10

We can set this up as a proportion

(1/5) gallon / (2/3) bin = x gallons / 1 bin      cross-multiply

1/5 = (2/3)x     multiply both sides by the reciprocal of 2/3 = 3/2

(3/2) (1/5) = x =  3/10 gallons fills the entire bin

Look at it this way......each 1/10 of a gallon fills 1/3 of the bin......so 3(1/10) gallons fills 3(1/3)bins....or

3/10 galons fills a whole bin....