Logarithmic growth

Example: Tree growth

Question

The height $H$ (in cm) of a certain sapling $t$ weeks after it was planted is modeled by $H=2+5\ln (t+1).$

1. Find the initial height of the sapling.
2. Find the height of the sapling after 10 weeks, giving your answer to 3 significant figures.
3. How long does it take for the sapling to reach a height of 20 cm? Give your answer to 1 decimal place.

Solution

When $t=0$: $$\begin{array}{cc}H& =2+5\ln (0+1)\\ & =2+5\ln 1\\ & =2+5(0)\\ & =2\text{ cm}∎\end{array}$$

When $t=10$: $$\begin{array}{cc}H& =2+5\ln (10+1)\\ & =2+5\ln 11\\ & \approx 14.0\text{ cm}∎\end{array}$$

The sapling reaches 20 cm when $H=20.$ $$\begin{array}{cc}20& =2+5\ln (t+1)\\ 18& =5\ln (t+1)\\ 3.6& =\ln (t+1)\\ t+1& ={\mathrm{e}}^{3.6}\\ t& ={\mathrm{e}}^{3.6}-1\\ & \approx 35.6\text{ weeks}∎\end{array}$$