If Emery has $1400 to invest at 11% per year compounded monthly, how long will it be before he has $2300. If the compounding is continuous, how long will it be?

(a) ##t = 4.52 years## or about 54 months
(b)##t=4.5## years
Given:
##P = 1400##
##r=0.11##
##m = 12 ( monthly)##
##A=2300##
##t=?## (no. of years)
(a) Formula:
##A=P(1+i)^n## where ##i=r/m## and ##n=mt##
Solution:
##2300=1400(1+0.11/12)^(12t)##
divide both sides by 1400
##(2300/1400)= 1.0092^(12t)##
get the logarithm of both sides
##log(2300/1400)= log(1.0092)^(12t)##
##0.2156=12t(3.9772×10^-3)##
dividing both sides of the equality by ##12(3.9772×10^-3)## gives
##t = 4.52 years## or about 54 months
(b) Formula:
##A = Pe^(rt)##
##2300=1400e^(0.11t)##
divide both sides by 1400
##2300/1400=e^(0.11t)##
get the natural logarithm of both sides
##ln(2300/1400)=lne^(0.11t)##
##0.49463=0.11t##
dividing both sides by 0.11,
##t=4.5## years