Question

Lydia invested $8,300 in an account paying an interest rate of 2.2% compounded

daily. Assuming no deposits or withdrawals are made, how long would it take, to the
nearest tenth of a year, for the value of the account to reach $12,690?

Answer 1

To calculate the time it takes for the value of the account to reach $12,690, we can use the formula P(1 + r/n)^(nt) = A. In this case, the principal amount is $8,300, the interest rate is 2.2%, and the final amount is $12,690. Solving the equation using logarithms will give us the approximate time, rounded to the nearest tenth of a year.

Step-by-step explanation:

To calculate the time it takes for the value of the account to reach $12,690, we can use the formula P(1 + r/n)^(nt) = A, where P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, t is the time in years, and A is the final amount. In this case, the principal amount is $8,300, the interest rate is 2.2% (or 0.022), and the final amount is $12,690.

Substituting these values into the formula, we have:

$8,300(1 + 0.022/365)^(365t) = $12,690

To solve for t, we can use logarithms. Taking the natural logarithm (ln) of both sides, we get:

ln($8,300(1 + 0.022/365)^(365t)) = ln($12,690)

Simplifying, we have:

ln($8,300) + ln(1 + 0.022/365)^(365t) = ln($12,690)

Next, we isolate t by subtracting ln($8,300) from both sides:

ln(1 + 0.022/365)^(365t) = ln($12,690) - ln($8,300)

Now, divide both sides by ln(1 + 0.022/365):

(365t) = (ln($12,690) - ln($8,300)) / ln(1 + 0.022/365)

Finally, divide both sides by 365 to solve for t:

t = [(ln($12,690) - ln($8,300)) / ln(1 + 0.022/365)] / 365