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Emplete the table for a savings account in which interest is compounded continuously. (Rounc
Annual
Time to
% Rate
Double
11 yr
Initial
Investment
1,000
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DETAILS
LARPRECALCRMRP7 3.5.016.
%
LABPRECALCRMBP735.018

Amount After
12 Years

Answer 1

Annual % rate = 6.3013% (4 d.p.)

Amount after 12 years = $2,130.07

Explanation:

To calculate the annual interest rate of a savings account where the interest is compounded continuously, we can use the continuous compounding formula.

`\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^(rt)$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}`

Given values:

• Initial investment, P = $1,000
• Time to Double, t = 11 years

When the investment is doubled, its value is 2P.

Therefore, to find the annual interest rate (r) based on the given information, substitute A = 2P and t = 11 into the formula, and solve for r:

`\begin{aligned}2P&=Pe^(11r)\\\\2&=e^(11r)\\\\\ln(2)&=\ln\left(e^(11r)\right)\\\\\ln(2)&=11r\ln\left(e\right)\\\\\ln(2)&=11r\\\\r&=(1)/(11)\ln(2)\\\\r&=0.06301338....\end{aligned}`

Therefore, the annual interest rate is 6.3013% (4 d.p.).

To find the amount (A) in the savings account after 12 years, substitute P = 1000, t = 12 and the found interest rate (r = 0.063013) into the formula and solve for A:

`\begin{aligned}A&=1000 \cdot e^(0.063013\cdot 12)\\\\A&=1000 \cdot e^(0.756156)\\\\A&=1000 \cdot 2.13007246....\\\\A&=2130.072464....\\\\A&=2130.07\; \sf (nearest\;cent)\end{aligned}`

Therefore, the amount in the savings account after 12 years will be $2,130.07 (rounded to the nearest cent).

`\hrulefill`

Note: We have used the rounded value of the interest rate in the calculation to find the amount in the account after 12 years. If we use the exact (unrounded) interest rate, the amount is $2,130.08.