Question

A person puts $100.00 into a savings account with 2.4% annual interest rate (computed continuously). The value of such an investment is given by: V=P\:e^{\left(rt\right)}\:V = P e ( r t ), where P is principal invested, r is the annual interest rate, and t is the number of years receiving interest. After how many years has the total interest exceeded $5.00? Round up to the nearest whole year.

Answer 1

It will take up to 3 years for the total interest to exceed $5.00

Step-by-step explanation:

The future value of an investment whose interest is compounded continuously can be expressed as;

A=P e^(rt)

where;

A=future value of the investment

P=initial value of investment

r=annual interest rate

t=number of years

In our case;

A=Initial value+interest=(100+5)=$105

P=$100

r=2.4%=2.4/100=0.024

t=unknown

replacing;

105=100 e^(0.024 t)

e^(0.024 t)=105/100

e^(0.024 t)=1.05

ln {e^(0.024t)}=ln 1.05

0.024 t ln e=ln 1.05

but ln e=1

0.024 t=ln 1.05

t=ln 1.05/0.024

t=2.03 years rounded up=3 year

It will take up to 3 years for the total interest to exceed $5.00