Alan

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This is the identity property of addition.

Kinetic energy is (1/2)*mass*velocity².

Work out (1/2)m1*v1² + (1/2)m2*v2² for the blocks before the collision and then after the collision.

If the energies are the same the collision was elastic; if energy after is less than energy before, it is inelastic; if energy after is greater than energy before, there is a gain in kinetic energy.

One way might be as follows:

The nearest square to 47 is 49; and 47 = 49 - 2 or 47 = 49*(1-2/49)

Now take the square root of this:

[49*(1-2/49)]^1/2 = 49^1/2*(1-2/49)^1/2 = 7*(1-2/49)^1/2.

Perform a binomial expansion of the term in brackets:  (1-2/49)^1/2 = 1 - (1/2)*2/49 + higher order terms.

The higher order terms get smaller and smaller, so drop them to get (1-2/49)^1/2 ≈ 1 - 1/49 = 48/49

Then 47^1/2 ≈ 7*48/49 

$${\frac{{\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{48}}}{{\mathtt{49}}}} = {\frac{{\mathtt{48}}}{{\mathtt{7}}}} = {\mathtt{6.857\: \!142\: \!857\: \!142\: \!857\: \!1}}$$

or 47^1/2 ≈ 6.86

Compare this with:

$${\sqrt{{\mathtt{47}}}} = {\mathtt{6.855\: \!654\: \!600\: \!401\: \!044\: \!1}}$$

The approximate value above agrees to 2 decimal places.

Volume = Mass/density.

Now plug the numbers in!

If n is the number of years for the populations to match then you want to solve:

50000*1.07ⁿ = 70000*0.96ⁿ for n

Take logs of both sides and use the fact that log(a*b) = log(a) + log(b) and log(aⁿ) = n*log(a)

log(50000) + n*log(1.07) = log(70000) + n*log(0.96)

$${log}_{10}\left({\mathtt{50\,000}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{n}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{1.07}}\right) = {log}_{10}\left({\mathtt{70\,000}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{n}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{0.96}}\right) \Rightarrow {\mathtt{n}} = {\frac{\left({\mathtt{35\,885\,121}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{70\,000}}\right)}{\mathtt{\,-\,}}{\mathtt{35\,885\,121}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{50\,000}}\right)}\right)}{{\mathtt{3\,892\,841}}}} \Rightarrow {\mathtt{n}} = {\mathtt{3.101\: \!679\: \!961\: \!830\: \!667\: \!4}}$$

so n ≈ 3years.

Subtract the second equation from the first and you get:

x - 5y - 3z = 2

But the third equation says that

x - 5y - 3z = 5

They can't both be true at the same time, hence there is no solution!

Volume = mass/density

The density of gold is about 19.3 gm/cm³ or 1205 lb/ft³ or 0.6973 lb/in³.

Can you take it from there?

49.3 milligrams is 49.3*10^-3 grams;

so number of atoms = 49.3*10^-3/(1.79*10^-22) = 49.3*10¹⁹/1.79 ≈ 2.75*10²⁰ atoms