Question

suppose you invested $1600 at an annual interest rate of 4.6% compounded continuously. how much will you have in the account after 4 years? round to the nearest cent.

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after $t$ years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (in decimal form), and $t$ is the time the money is invested for in years.

Step2: Identify the values of P, r, and t

Given that $P=\$1600$, $r = 4.6\%=0.046$ (converted from percentage to decimal by dividing by 100), and $t = 4$ years.

Step3: Substitute the values into the formula

Substitute $P = 1600$, $r=0.046$, and $t = 4$ into the formula $A=Pe^{rt}$. So we have $A=1600\times e^{0.046\times4}$.
First, calculate the exponent: $0.046\times4 = 0.184$. Then we need to calculate $e^{0.184}$. We know that $e\approx2.71828$, so $e^{0.184}\approx2.71828^{0.184}$. Using a calculator, $e^{0.184}\approx1.20209$.
Then, multiply this by the principal: $A = 1600\times1.20209$.

Step4: Calculate the final amount

$1600\times1.20209 = 1923.344$. Rounding to the nearest cent (two decimal places), we get $A\approx\$1923.34$.

Answer:

$\$1923.34$