Alan

Glad to have been of assistance.  

My own knowledge of electricity and electronics doesn't extend much beyond Ohm's law and Kirchoff's laws, but as long as there are some mathematical equations in there somewhere I'm happy!

Glad to note that you are making an effort to learn algebra.  A little effort goes a long way! 

The radioactive decay equation is N = N0e^-ln(2)*t/τ, where τ is the half-life, N0 is the number of atoms at time t = 0 and N is the number at time t.

Rewrite this as (N/N0) = e^-ln(2)*t/τ, where, here, N/N0 is 0.65  (i.e. 65% expressed as a fraction). So:

0.65 =  e^{-ln(2)*t/5730}.

Take logs of both sides:

ln(0.65) = -ln(2)*t/5730.

Rearrange:

t = -5730*ln(0.65)/ln(2) years

$${\mathtt{t}} = {\mathtt{\,-\,}}{\frac{{\mathtt{5\,730}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{0.65}}\right)}}{{ln}{\left({\mathtt{2}}\right)}}} \Rightarrow {\mathtt{t}} = {\mathtt{3\,561.128\: \!398\: \!756\: \!128\: \!174\: \!2}}$$

or the artifact was made ≈ 3560 years ago.

As a rough approximation the temperature will fall exponentially.  That is the temperature as a function of time is given by:

T(t) = Ta+(T0-Ta)*e^-k*t

T(t) is the temperature at time t

T0 is the temperature at time zero (185°F)

Ta is the ambient temperature (51°F)

k is a constant we can find from knowing that the temperature at t=30mins is 150°F

150 = 51 + (185-51)*e^-k*30

ln((150-51)/(185-51)) = ln(e^-k*30)

ln(99/134)=-k*30

k = -ln(99/134)/30  ≈ 0.01 per minute.

So the temperature after 45 minutes is:

T(45) = 51 + 134*e^-0.01*45 ≈  136.4°F

Time for turkey to cool to 100°F is given by:

100 = 51 + 134*e^-0.01*t

ln(49/134) = -0.01*t

t = -100*ln(49/134) ≈ 100.6 minutes

In the first rearrangement of the first equation: where are you getting the 5 from? 

(50-V2)/10 + (V3-V2)/20 + (0-V2)/15 = 0 This can be written as:

50/10 - V2/10 +V3/20 - V2/20 +0/15 - V2/15 = 0

The first term above is just 50/10 = 5; so that's where the 5 comes from.

Same with the bit where  we have 5 - 13/60 where is the 13/60 coming from?

Collect all the terms involving V2:  -V2/10 - V2/20 - V2/15  Factor out -V2 to get:

-V2*(1/10 + 1/20 + 1/15)  Collect the fractions together as a single fraction.  To do this we need to find a common multiple of 10, 20 and 15.  Any multiple will do, but often life is easier if we use the lowest common multiple.  In this case that is 60, so we can express the individual fractions as:

1/10 = 6/60

1/20 = 3/60

1/15 = 4/60

So now the sum can be written as 6/60 + 3/60 + 4/60, and because they are all over the same number we can write  6/60 + 3/60 + 4/60 = (6 + 3 + 4)/60 = 13/60.

Hope that helps!

Shelley's score = 90x

Terrence's score = 90x-20

Difference between them is (90x - 20 - 90x) = -20

So the constant term represents the difference between Shelly's and Terrence's scores.

Melody, if you put A = 2P (investment doubled) then take logs you get ln(2) = r*t or t = ln(2)/r.

The calculator on the Home page gives something slightly different:

It interprets the first part (up to and including the 2) as computerese for 1.0*10^-2 and then gets stuck because it expects an operator before the decimal point!

A number like 9.0e9 may be interpreted as 9.0*10⁹ only when the value following the "e" is an integer (positive or negative) and there is no exponentiation symbol ("^") between it and the "e".  In the question here, these conditions don't hold so we have to interpret this as:

$${\mathtt{10\,000}}{\mathtt{\,\times\,}}{{\mathtt{e}}}^{-{\mathtt{2.835}}} = {\mathtt{587.185\: \!258\: \!338\: \!722\: \!9}}$$ 

No.  Everything on the right-hand side would have to be over the same denominator before you could flip it.

So, here, for example you could collect everything on the RHS over 42.3*10²,

1/x = (9.4+4.5)/(42.3*10²) 

1/x = 13.9/(42.3*10²)  Now you can invert:

x = 42.3*10²/13.9  ≈ 304.32