You want a savings account to grow to $5000 in 10 years. If the interest rate is 3.5% compounded continuously, how much should you invest?

To find out how much you should invest in order to grow your savings account to $5000 in 10 years with an interest rate of 3.5% compounded continuously, you can use the formula for continuous compound interest:

A = P * e^(rt)

In this formula:

A represents the final amount or value of the investment (in this case, $5000).

P represents the principal or initial investment amount (which is what you're trying to find).

e represents the mathematical constant (approximately 2.71828).

r represents the interest rate (3.5% or 0.035 when expressed as a decimal).

t represents the number of years (10).

Now we can plug in the given values into the formula and solve for P:

5000 = P * e^(0.035 * 10)

To isolate P, divide both sides of the equation by e^(0.035 * 10):

5000 / e^(0.035 * 10) = P

Using a calculator, you can calculate the value of e^(0.035 * 10), which is approximately 1.4267. So:

P ≈ 5000 / 1.4267

P ≈ 3503.50

Therefore, you should invest approximately $3503.50 in order to grow your savings account to $5000 in 10 years with a continuous interest rate of 3.5%.

Continuous compounding is a method where interest is calculated and added to the investment continuously over time. This means that the interest is compounded infinitely many times within a specific time period (in this case, 10 years). Continuous compounding offers higher growth compared to other compounding frequencies, such as annually or semi-annually. The formula used for continuous compound interest is a derived formula from the exponential function, utilizing the mathematical constant e (approximately 2.71828).