Question

MODELING EXPONENTIAL GROWTH

1. You put $3,800 dollars in a savings account. The bank will provide 1.8% interest every year. Write and solve a model that describes how much money will be in the account in 15 years.

2. Suppose you deposit $2000 into a savings account that pays an interest annual rate of 4% if no money is added or withdrawn from the account, how much will be in the account after 3 years? What about 18 years? How many years will it take for the account to contain 3000 (for this now you know the total amount, you need to find t time)?

Answer 1

1. M = C*1.018^t

After 15 years: M = 4965.93

2. After 3 years: M = 2249.728

After 18 years: M = 4051.633

Time to achieve 3000: t = 10.338 years

Explanation:

1. Since the money gets increased every year at a rate of 1.8% after one year it'll be the initial amount multiplied by 1.018, so:

After one year:

M = C*1.018

After two years:

M = C*1.018*1.018 = C*(1.018)²

After three years:

M = C*(1.018)²*1.018 = C*(1.018)³

And so on, therefore:

M = C*(1.018)^t

Where M is the final amount, C is the initial amount and t is the time elapsed in years. For this case we have:

M = 3800(1.018)^15 = 4965.92626

2. Applying the same line of thought as above, we have:

M = C*(1.04)^t

After 3 years:

M = 2000*(1.04)^3 = 2249.728

After 18 years:

M = 2000*(1.04)^18 = 4051.633

To obtain 3000:

3000 = 2000*(1.04)^t

2000*(1.04)^t = 3000

1.04^t = 3000/2000

1.04^t =1.5

log(1.04^t) = log(1.5)

t*log(1.04) = log(1.5)

t = log(1.5)/log(1.04) = 10.338 years