There are 500 seeds in an average pumpkin. It takes 20 weeks to produce a vine with 1 pumpkin from a seed and then the vine withers. A live vine covers 2 square meters of land. The earths diameter is 12,742 km and its surface is 33% land.
I'm a diabolical psychopath and I have managed to obtain 1 pumpkin seed! How long will it take me to cover all land mass with pumpkin vines?
On
We can find the surface area of the earth $$SA = \frac{4}{3}\pi r^2=\frac{4}{3}\pi (\frac{12,742}{2}km)^2=\frac{4}{3}\pi (6371km)^2=\frac{4}{3}\pi (6371000m)^2$$ Scaled to $\frac{1}{3}$ (Assuming $33\% = \frac{1}{3}$) $$SA=4\pi(6371000)^2$$ Knowing each plant takes $2m^2$ of land, we will need $$Plants = \frac{SA}{2}=2\pi(6371000)^2$$ We will get pumpkin seeds at a rate of $$Seeds = (500)^{n}$$ where $n=20 \text{ weeks}$.Therefore we need to solve $2\pi(6371000)^2=500^n$ for $n$. $$2\pi(6371000)^2=500^n \\ \log(2\pi(6371000)^2)=\log(500^n) \\ \log(2\pi(6371000)^2)=n\log(500) \\ n= \frac{\log(2\pi(6371000)^2)}{\log(500)} \\ n \approx 5.34$$ Note that $n$ is in units of $\frac{1}{20 \text{ weeks}}$ so $$n \approx 106.76 \text{ weeks} \\ n \approx 747.29 \text{ days}$$
Thinking about weeks and seeds, pumpkins and vine covered area, we come up with \begin{align} p(0) &= (1, 0, 0) \\ p(20) &= (500, 1, 2) \\ p(40) &= (500^2, 500, 1000) \end{align} then we laugh madly, BWAHAHAHAHHHAHAA, and note $$ p(20 k) = (500^k, 500^{k-1}, 2 \cdot 500^{k-1}) \quad (k \ge 1) $$ The feasible area is $$ A = p \, 4\pi r^2 = p \, 4 \pi (d/2)^2 = p \pi d^2 = 2 \cdot 500^{k-1} $$ Taking a logarithm gives: $$ \ln(p\pi d^2) = \ln(2) + (k-1) \ln(500) \Rightarrow \\ k = 1 + \frac{\ln(p\pi d^2) - \ln(2)}{\ln(500)} = 6.159415997965432 $$ $20$ weeks, where $p = 33 \%$ and $d = 12742000$m.
So we need $7$ cycles of $20$ weeks of mad fun.