Answer: When r= 9.5% , time = 7.3 years
When r= 2.5% , time = 27.7 years
Step-by-step explanation:
Exponential equation for compounded continuously:
[tex]A=Pe^{rx}[/tex] , where P= Principal value invested , r= rate of interest, x= time
Given : P= $6,000
A = 2P= 2 ($6,000) = $12,000 (i)
When r= 9.5%= 0.095
[tex]A= 6000e^{0.095x}[/tex] (ii)
From (i) and (ii)
[tex]6000e^{0.095x}=12000\\\\\Rightarrow\ e^{0.095x}=2\\\\\Rightarrow\ \ln e^{0.095x} =\ln 2\\\\\Rightarrow\ 0.095x=0.693147\\\\\Rightarrow\ x=\dfrac{0.693147}{0.095}=7.296284210\approx7.3 \text{ years}[/tex]
For r= 2.5% = 0.025
[tex]6000e^{0.025x}=12000\\\\\Rightarrow\ e^{0.025}=2\\\\\Rightarrow\ \ln e^{0.025x} =\ln 2\\\\\Rightarrow\ 0.025x=0.693147\\\\\Rightarrow\ x=\dfrac{0.693147}{0.025}=27.72588\approx27.7 \text{ years}[/tex]
