Question

suppose that $32,000 is deposited into an account and the balance increases to $34,665.19 after 2 years. How long will it take for the account to grow $57,382.30. Assuming continuous compounding.

It will take about _ years for $32,000 to grow to $57,382.30

Answer 1

Answer :

Data given:

Initial investment or principal amount, P = $32000

Final amount after 2 years in the account, A = $34665.19

Rate of interest, r = ?

Time period, t = 2 years

Interest is continuously compounded on the investment.

As we know, the formula for calculating the final amount after a period of continuous compounding is given as -

A= Pent

where P - initial investment or principal amount

r - rate of interest

t - time period of investment

A - final amount

Using the above formula, we can compute the required rate of interest as -

A= Pent

21 34665.19 32000 e

$\ln \frac{34665.19 } {32000} = 2r$   $(From \ the \ natural \ logarithm \ property: \ \ \ a= e^{b} \ \ \Rightarrow \ \ \ln a = b)$

$\frac{1}{2} \ln \frac{34665.19 } {32000} = r$

$\therefore \ r= 0.04 \ \ \ or \ \ \ 4\%$

Now, using the above information, we can find the time required for the investment to grow from $32000 to $57382.30 as -

A= Pent

$\frac{57382.30} {32000} = e^{0.04 t}$

$\ln \frac{57382.30} {32000} = 0.04 t$$(From \ the \ natural \ logarithm \ property: \ \ \ a= e^{b} \ \ \Rightarrow \ \ \ln a = b)$

$\frac{1}{(0.04)} \ln \frac{57382.30} {32000} = t$

Thus, it will take about 14.6 years for $32,000 to grow to $57,382.30 .