Alan

Rate of growth is dN/dt = -0.412/100*N which can be solved to get  N_t = N₁₂*e^-0.412/100*t where N₁₂ = 48810427.  (I've assumed the -0.412% means -0.412% per year)

So, to find N₁₁ (the number  in 2011) we set t = -1 (i.e. it's 1 year earlier) to get:

$${\mathtt{N11}} = {\mathtt{48\,810\,427}}{\mathtt{\,\times\,}}{{\mathtt{e}}}^{{\mathtt{\,-\,}}\left({\frac{{\mathtt{0.412}}}{{\mathtt{100}}}}{\mathtt{\,\times\,}}\left(-{\mathtt{1}}\right)\right)} \Rightarrow {\mathtt{N11}} = {\mathtt{49\,011\,940.792\: \!604\: \!870\: \!237\: \!79}}$$

or N₁₁ ≈ 49011941

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The component of the 2kN force along AC is zero, as it is perpendicular to AC!

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cosine only takes values between ±1, so there is no angle whose cosine is 1.21

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If you write this equation in the form y = x + 4, then you can see that its slope is 1 (the coefficient of the x term).  Any line perpendicular to this will have a slope of -1  (= the negative inverse of the slope of the given line).  

However, there will be an infinite number of lines that are perpendicular to the given line, so, a general equation for the perpendicular line is y = -x + c where c is a constant.  You need more information (such as a point through which you know the perpendicular line goes) before you can specify c.

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The radioactive decay equation is N/N0 = e^-ln(2)*t/τ, where N/N0 is the fraction at time t, and τ is the half-life. If we want to find t, we can take logs of both sides and rearrange to get:  t = -ln(N/N0)*τ/ln(2)

U238 has τ = 4.468*10⁹ years, so here we have 

$${\mathtt{t}} = {\mathtt{\,-\,}}{\frac{{ln}{\left({\mathtt{0.65}}\right)}{\mathtt{\,\times\,}}{\mathtt{4.468}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{{\mathtt{9}}}}{{ln}{\left({\mathtt{2}}\right)}}} \Rightarrow {\mathtt{t}} = {\mathtt{2\,776\,810\,067.302\: \!335\: \!197\: \!609\: \!348\: \!8}}$$

or t ≈ 2776810067 years.

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Depends what is meant by "distinct" I guess.  If you reflect Melody's triangle in any of its sides, you get another one that can't be rotated into the first.

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kes, you have written "Rational number is a number that is expressed in the form pq , where p and q are Integers and q ≠ 0."

This is incorrect.  It should read: "Rational number is a number that is expressed in the form p/q , where p and q are Integers and q ≠ 0."  Note the "p/q" not "pq".

This comment affects some of your other statements as well.  I recommend that you edit your note to correct these.

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Your answer is the same as heureka's, sabi92; it just looks slightly different!  Multiply the top and bottom of your expression by -1 to get it to look like heureka's.

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Note from the original diagram that there is a right-angle sign between the force arrow and the beam AC, and just to make Melody's reply clearer, here is the diagram with the 'X' that Melody used marked on it.

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If we assume you meant out of 125 beads, 113 were gold and 12 were silver, then (assuming there were only gold and silver beads in the container):

Gold/5000 = 113/125 so Gold = 5000*113/125 = 4520

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