Question

Meena is going to invest in an account paying an interest rate of 3% compounded continuously. How much would Meena need to invest, to the nearest hundred dollars, for the value of the account to reach $470 in 13 years?

Answer 1

The amount Meena would need to invest to the nearest hundred dollars is $300.

Explanation:

Use the continuous compounding interest formula to calculate how much Meena would need to invest in an account paying an interest rate of 3% compounded continuously, for the value of the account to reach $470 in 13 years.

`\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:

• A = $470
• r = 3% = 0.03
• t = 13 years

Substitute the given values into the continuous compounding interest formula and solve for P:

`\implies 470=Pe^((0.03 * 13))`

`\implies 470=Pe^(0.39)`

`\implies P=(470)/(e^(0.39))`

`\implies P=(470)/(1.47698079...)`

`\implies P=318.216731...`

`\implies P=\$318.22`

Therefore, the principal that would need to be invested for the value of the account to reach $470 in 13 years is $318.22.

$318.22 rounded to the nearest hundred dollars is $300.

Therefore, the amount Meena would need to invest to the nearest hundred dollars is $300.