Alan

You have it upside-down Anonymous!  The slope is (y2 - y1)/(x2 - x1), so (6 - 8)/(6 - (-4)) = -2/10 = -1/5.

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First use sin^-1(5/12)  (or asin(5/12) which is the same thing).  This will give an angle in the first quadrant. 

Next subtract this angle from 180°, to give an angle whose sine is also 5/12.

Then use cos and tan on this angle (you should find that both cos and tan values are negative).

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What is the given event?!!!!!

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The ball will displace its own volume of water, namely (4/3)*pi*4^3 cubic inches.

This will appear as an increase in height, h, of the water in the tank such that pi*(20^2)/4*h =  (4/3)*pi*4^3

or h = (4/3)*4^4/20^2 in.

$${\mathtt{h}} = {\frac{\left({\frac{{\mathtt{4}}}{{\mathtt{3}}}}\right){\mathtt{\,\times\,}}{{\mathtt{4}}}^{{\mathtt{4}}}}{{{\mathtt{20}}}^{{\mathtt{2}}}}} \Rightarrow {\mathtt{h}} = {\mathtt{0.853\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

so h ≈ 0.853 in.

(Note: The cross-sectional area of the tank is given by pi*diameter^2/4).

With a quadratic equation there are always two roots (though one might be repeated), so, if x1 and x2 are the roots, then the equation could be expressed as y = (x - x1)*(x - x2)  or y = -(x - x1)*(x - x2).  You would need extra information to determine which of these two it is (one is concave up, the other is concave down).

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Never mind the whole works, just consider the phrase "to be or not to be" There are 18 characters (including the spaces).  Suppose the monkeys can type 18 characters a second and they only ever hit the 26 lower case letters of the English alphabet plus the space key.  Then there are 27^18 possibilities, so this many/18 seconds to get through them all (hence to guarantee getting the phrase). This many seconds is:

$${\frac{{{\mathtt{27}}}^{{\mathtt{18}}}}{\left({\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{3\,600}}{\mathtt{\,\times\,}}{\mathtt{24}}{\mathtt{\,\times\,}}{\mathtt{365}}{\mathtt{\,\times\,}}{\mathtt{1\,000}}\right)}} = {\mathtt{102\,439\,781\,348\,723.257\: \!530\: \!001\: \!284\: \!246\: \!6}}$$

Approximately 102439781348723 millenia.

Don't hold your breath!

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If tan²(x) = 1/4 then tan(x) = ±1/2 so x = tan^-1(±1/2)

Use atan on the calculator on the home page here to find the values of x (atan is the same as tan^-1).  This will give you angles in the first and fourth quadrants, but there will also be solutions in the second and third quadrants (obtained by 180° ± x).

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The thousands column is the middle column!

    1       2       3                4                  5                          6                                7

units  tens  hundreds  thousands  ten-thousands  hundred-thousands   millions

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You are right about the probability of the thousands digit equalling 5 being 1/3 Melody.  I just checked my program and realised I'd put the counter in the wrong place!  

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If log10 = log(2x-3) then 10 = 2x-3 so 2x = 13 and x = 13/2

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