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2.4 Continuous Growth and Decay: Worked Examples

Example 1.  Write a formula for an account giving 15% annual interest compounded continuously with initial value of $2,200.

SOLUTION.   The general continuous growth function is of the form , with  P  as the initial value and r  as the annual interest rate, so the function is given by

Example 2.  How much money would be in an account after 25 years if you deposited $5,000 at each of the following annual interest rates compounded continuously?

a)     11%                                                 b) 2.8%

c)     5.6%                                               d) 7.45%

SOLUTION.  

a)     

b)     

c)     

d)     

Example 3.  Which is the better deal:  An account that pays 5% interest compounded monthly or one that pays 4.95% compounded continuously?

SOLUTION.   Find the effective yield for compounding monthly by using the compound interest formula with t = 1 and P = 1:

The effective yield is . Now, find the effective yield for continuous compounding by using the continuous compounding formula with t = 1 and P = 1:

The effective yield is . A better deal.

Example 4.  Find much money would you need to invest in an account that gives 8.1% interest compounded continuously if you want to have $100,000 in this account in 20 years?

SOLUTION.   Use the continuous compounding formula with t = 20 and r =0.081:

Solve for P:

Example 5.  Given that an amount (in grams) of Radium-226 decays in such a way that after t  years, the amount left is given by

a)     What is the initial amount of Radium-226?

b)     How much Radium-226 remains after 2,000 years?

SOLUTION.  

a)    Using t=0:

b)    Using t=2,000:

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