Suppose that ?$19,171 is invested at an interest rate of 6.5?% per? year, compounded continuously.

a) Find the exponential function that describes the amount in the account after time? t, in years.

?b) What is the balance after 1? year?

2? years?

5? years?

10? years?

?c) What is the doubling? time?

 

a. A = ($19,171)e^0.065t

b. A_1year = $20,458.50566

A_2year = $21,832.47894

A_5year = $26,533.25151

A_10year = $36,722.83323

c. t = 10.66 years

Step-by-step explanation

Hello! To help solve this problem, you should be able to familiarize yourself with the concept of exponential functions. For this problem, we will use the general formula for continuously compounding interest.

A = Pe^rt

where,

A = Future Value of the amount

P = Present Value of the amount

r = interest rate

t = years passed (time)

Now, the problem give us the following data; ?$19,171 is invested at an interest rate of 6.5?% per? year. Therefore, we can derive the function

A = ($19,171)e^0.065t

Next, to find the balance after a certain amount of time, we simply substitute t. Therefore we have

@ 1 year

A_1year= ($19,171)e^0.065t ; t=1

A_1year= ($19,171)e^0.065(1)

A_1year = $20,458.50566

@ 2 years

A_2year= ($19,171)e^0.065t ; t=2

A_2year= ($19,171)e^0.065(2)

A_2year = $21,832.47894

@ 5 years

A_5year= ($19,171)e^0.065t ; t=5

A_5year= ($19,171)e^0.065(5)

A_5year = $26,533.25151

@ 10 years

A_10year= ($19,171)e^0.065t ; t=10

A_10year= ($19,171)e^0.065(10)

A_10year = $36,722.83323

Next, to get the doubling time, we simply need to assume that A = $38,342 and equate our equation to get the value of t.

A = ($19,171)e^0.065t

$38,342 = ($19,171)e^0.065t

2 = e^0.065t

t = 10.66 years

I hope you've umderstood something from my explanation! Feel free to ask me for some clarification if you have something you don't understand.