Exponential Equations 15

Question

Find the time it takes for $6,100 to double when invested at an annual interest rate of 10%, compounded continuously.

how many years  ?

Find the time it takes for $610,000 to double when invested at an annual interest rate of 10%, compounded continuously.

how many years  ?

Answer 1

Since this is a continual compounding situation, the formula we will use involves the natural number, e, and is this: A = Pe^rt where A is the final amount, P is the initial amount of the deposit, r is the interest rate, and t is the time in years. Since we want the amount of $6,100 to double, the P value is 6,100 and the A value is 12,200. The r value, written as a decimal, is .1, and the t value is our only unknown. So we insert the values into the formula and we get:

12,200 = 6,100e^.1t.

To solve this, we first need to isolate e^.1t, so we divide both sides by 6,100, so we get 2 = e^.1t. At this point, we need to isolate the exponent containing the variable, so we take the natural log of both sides:

ln 2 = ln e^.1t. Since the natural log of e raised to any exponent is just the exponent, the right side of the equation is simply .1t. For the left side, we can use a calculator to find ln 2 which is .693147. So the equation is now .693147 = .1t. To finish, we divide both sides by .1 and we get t = 6.93, so in about 7 years the money should double.