Naomi invested $920 in an account paying an interest rate of 4.7% compounded continuously. Assuming no deposits or withdrawals are made, how long would it take, to the nearest year, for the value of the account to reach $2,310?

If the initial amount (also called as principal amount) is P, and the interest rate is R% per unit time, and it is left for T unit of time for that compound interest, then the interest amount earned is given by:

[tex]CI = P(1 +\dfrac{R}{100})^T - P[/tex]

The final amount becomes:

[tex]A = CI + P\\A = P(1 +\dfrac{R}{100})^T[/tex]

For this case, we're provided that:

Initial amount invested = P = $920
Rate of interest = 4.7% (assumingly annual rate) = R
Type of interest = Compound interest
Final amount after some time = $2310 = A
The value of that "some time" is to be known.

Let the time be 'T' years after which this investment becomes $2310

Then, by the formula stated above, we get:

[tex]A = P\left(1 +\dfrac{R}{100}\right)^T\\\\2310 = 920\left(1 +\dfrac{4.7}{100}\right)^T\\\\\\2310 = 920\times(1.047)^T\\\\(1.047)^T = \dfrac{2310}{920}\\\\\\T = \log_{1.047}\left( \dfrac{2310}{920} \right)\\[/tex]

Using calculator, we get:

[tex]T \approx 20.0446 \approx 20 \: \rm years[/tex] (as unit of time in rate of interest was assumed to be yearly)

Thus, the time that it will take, to the nearest year, for the value of the account to reach $2,310 for this case is 20 years.

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