Alan

Write this as (x⁴)^1/5(x³)^1/4(x²)^1/3(x)^1/2

This is x^{(4/5+3/4+2/3+1/2)}

4/5 + 3/4 + 2/3 + 1/2 = 163/60 

so x^{(4/5+3/4+2/3+1/2)} = x^163/60 

The vertical lines either side of a varaible or number mean take the absolute value, they are not lower-case l's.

For example, if x = -7 then |x| is just 7.

Well spotted Chris!

Use mod(x,y) or x mod y (It can be entered in either format).  For example:

$${\mathtt{13}} \,{mod}\, {\mathtt{4}} = {\mathtt{1}}$$

Looks like f(x) = -1- x²

               g(x) = sin(pi*x/2)

It depends on what information you have.  If you have two (x, y) points then it is the difference between the y values divided by the difference between the corresponding x values.

Set W(t) = 10³ and solve for t.

10³ = 6*10¹⁰*e^-0.000122t  You didn't have an "e" in your expression, but I suspect it should be there, so I've included it.

Divide both sides by 6*10¹⁰

10^-7/6 = e^-0.000122t  

Take logs (natural log) of both sides:

ln(10^-7/6) = -0.000122t

Divide both sides by -0.000122

t = -ln(10^-7/6)/0.000122 = (7*ln(10)+ln(6))/0.000122

$${\mathtt{t}} = {\frac{\left({\mathtt{7}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{10}}\right)}{\mathtt{\,\small\textbf+\,}}{ln}{\left({\mathtt{6}}\right)}\right)}{{\mathtt{0.000\: \!122}}}} \Rightarrow {\mathtt{t}} = {\mathtt{146\,802.091\: \!149\: \!068\: \!645\: \!811}}$$

t ≈ 146802 years

If you start timing from the instant the first apple falls off then the time is 170*61mins because there are only another 61 left to fall!

$${\mathtt{t}} = {\mathtt{170}}{\mathtt{\,\times\,}}{\mathtt{61}} \Rightarrow {\mathtt{t}} = {\mathtt{10\,370}}$$ minutes

t ≈ 7.2 days

Set N = 4N₀.

4 = e^0.017t

Take logs of both sides (use natural logs)

ln(4) = 0.017*t

Divide both sides by 0.017

t = ln(4)/0.017

$${\mathtt{t}} = {\frac{{ln}{\left({\mathtt{4}}\right)}}{{\mathtt{0.017}}}} \Rightarrow {\mathtt{t}} = {\mathtt{81.546\: \!727\: \!124\: \!699\: \!448\: \!2}}$$

t = 82 days to the nearest day