height=height at first(1+rate)^n
n=years
The height of this oak will be:
Data:
height at first=60 cm
rate=8%=8/100=0.08
height=60(1+0.08)^n=60(1.08)^n
The height of this conifer will be:
Data:
height at first=50 cm
rate=15%=15/100=0.15
height=50(1+0.15)^n=50(1.15)^n
The heights of both trees will be the same at n years, therefore:
60(1.08)^n=50(1.15)^n
Now, we can solve this logarithmic equation.
Ln[60(1.08)^n]=Ln[50(1.15)^n]
Ln 60+nLn 1.08=Ln 50+nLn 1.15
nln 1.08-nLn 1.15=Ln 50-Ln 60
n(ln 1.08-ln 1.15)=Ln 50-Ln 60
n=(Ln 50 - Ln 60) / (Ln1.08 - Ln 1.15)
n=2.90316784...≈2.9
Answer: 2.9 years.
