Final answer:
To find the interest rate required for a $500 investment to double in 4 years when compounded continuously, use the continuous compounding formula and solve for the rate, which involves division and taking the natural logarithm.
Step-by-step explanation:
To calculate the interest rate needed for a $500 investment to double in 4 years when compounded continuously, we can use the formula for continuous compounding, Pert = A, where P is the principal amount, r is the rate, t is the time in years, A is the amount of money accumulated after n years, including interest, and e is the base of the natural logarithm (approximately equal to 2.71828).
In this case, we want the investment to grow from $500 to $1,000, which means A is $1,000, P is $500, and t is 4 years. Substituting these values into the equation gives us 500e4r = 1,000. To find the rate, we can divide both sides by 500, resulting in e4r = 2. Taking the natural logarithm of both sides gives us 4r = ln(2), and dividing by 4 yields r = ln(2)/4. This calculation results in an interest rate, rounded to the nearest thousandth, that is required for the investment to double in 4 years when compounded continuously.