Find the missing values assuming continuously compounded interest. How to find the initial investment and annual % rate, with the given 5 years to double & $2500 amount after 10 years

Question

I'm having a hard time starting it and figuring out what formula to take. if someone can help me understand how to go about this, I should be able to take care of the other answers.

Answer 1

A=Pe^rt where P = initial amount invested, r=annual interest rate, t= number of years

2P = Pe^5r

2= e^5r

take natural logs of both sides

ln2 = 5r

r = ln2/5 = about .69/5 = about .1386 = 13.86% annual growth rate rounded off to 2 decimal places

2500=Pe^10r =Pe^1.386

2500/P = e^1.386

ln(2500/P) = 1.386

ln2500-lnP = 1.386

lnP = ln2500-1.386

lnP = 6.438

P = e^6.438 = 625

P =$625 rounded off to nearest dollar

or far more simply:

2500 doubled from 1250 in the 2nd 5 years

1250 doubled form 625 in the 1st 5 years

in 15 years, it will grow to 5000

2500/4 = 625

original amount invested = exactly $625

Answer 2

Here's another approach to Part II

Need $2,500 after 10 yrs @ 13.86% continuous compounding.

2500 = Pe^10(0.1386)

2500 = Pe^1.386

2500 = P(3.999)

P = 2500/3.999 = $625.15