Question

PLEASE HELP, I don't understand this question.

Michael has recently opened a savings account. The banker gave him an equation to predict his balance at any given month, assuming he made no further deposits nor withdrawals. The equation is as follows: b=975(1.003)^m


where "b" is the balance of the account, and "m" represents the number of months the account has been open.

a) How much did Michael deposit when he first opened the account?

b) Is this a case of exponential growth or exponential decay? How do you know?

c) What is the monthly interest rate on the account?

d) What is the account balance after 6 months?

...after 12 months?

e) How many months will it take to have at least $1000 in the account?


Thank you in advance!

Answer 1

a) Michael did not make any initial deposit since the equation only models the balance over time assuming no deposits nor withdrawals.

b) This is a case of exponential growth since the coefficient (1.003) is greater than 1. In an exponential growth model, the base (1.003 in this case) represents the factor by which the quantity being modeled grows over a given period.

c) The monthly interest rate can be determined by calculating the difference in the balance between two consecutive months and dividing by the previous month's balance. Let's use months 1 and 2 as an example:

b(1) = 975(1.003)^1 = 978.23
b(2) = 975(1.003)^2 = 981.48

The difference between the two balances is 981.48 - 978.23 = 3.25. To get the monthly interest rate, we divide this difference by the previous month's balance: 3.25/978.23 = 0.00332 or 0.332%.

d) After 6 months:

b(6) = 975(1.003)^6 = 1015.10

After 12 months:

b(12) = 975(1.003)^12 = 1056.08

e) We can solve for the number of months (m) it takes to reach a balance of $1000 by setting the equation equal to 1000 and solving for m:

1000 = 975(1.003)^m

Dividing both sides by 975:

1.0256 = 1.003^m

Taking the natural logarithm of both sides:

ln(1.0256) = m*ln(1.003)

Dividing both sides by ln(1.003):

m = ln(1.0256)/ln(1.003) ≈ 27.3

So, it will take approximately 28 months to have at least $1000 in the account.