Question

You have invested money in a savings account that pays a fixed monthly interest on the account balance. The following table shows the account balance over the first 5 months.

Time in months	Savings balance
0	$1500.00
1	$1521.00
2	$1542.29
3	$1563.88
4	$1585.77
5	$1607.97

(a) How much money was originally invested?
$  

(b) Show that the data are exponential. (Round your answer to three decimal places.)

Each successive ratio of new/old is   , which shows that the data is exponential.

Find an exponential model for the account balance. (Let t be the time in months and B the savings balance in dollars. Round your parameters to three decimal places.)
B(t) =

(c) What is the monthly interest rate? (Round your answer to one decimal place.)
  %

(d) What is the yearly interest rate? (Round your answer to one decimal place.)
%

(e) Suppose that you made this investment on the occasion of the birth of your daughter. Your plan is to leave the money in the account until she starts college at age 19. How large a college fund will she have? (Round your answer to the nearest cent.)
$

(f) How long does it take your money to double in value? (Round your answer to two decimal places.)
months

How much longer does it take it to double in value again? (Round your answer to two decimal places.)
months

Answer 1

A) Money originally inveeted is $1,500.

B) Exponential data is data which follows some patterns. I.e. variable moves upward or downward by common ratio or amount or function.

Each successive ratio of new/old is 1.014, which shows that the data is exponential.

Trend Ratio = (Amount ⁿ / Amount ⁿ-¹) * 100

1. ($1,521.00/$1,500.00) = 1.014

2. ($1,542.29/$1,521.00) = 1.014

3. ($1,563.88/$1,542.29) = 1.014

4. ($1,585.77/$1,563.88) = 1.014

5. ($1,607.97/$1,585.77) = 1.014

Exponential Model For Account Balance B(t) = A(1.014)t

(t= months)

C) MONTHLY INTEREST RATE

Monthly Interest Rate will be 1.4%

(1.014-1)*100 = 1.4% per month

D) ANNUITY INTEREST RATE

Nominal rate: 1.4% per month * 12 months

= 16.8% per annum coumpounded monthly

Effective rate: {(1.014)¹² -1}*100

= 18.16% per annum compounded annually

E) Balance after 19 years:

effective annual rate = 18.16% ( calculated in D)

Duration = 19 years

Balance after 19 years

= $1,500(1.1816)19 = $35,729.79

= $35,700.

F) Duration to double the money

Monthly rate is 1.4 i.e. r=0.014

3000 = 1500(1.014)t

2 = 1.014 t

Solving this equation using log table

t = Log(2)/Log(1.014)

t = 0.3010/0.0060 = 50.16 i.e 50 months approx.

Dear student, please refer the above solution and provide feedback as well as query if any are invited by comment.