Applications of Exponential Functions: Practice Exercises

Here are various Exercises to accompany the section Applications of Exponential Functions.  

The Origins of Exponential Functions

  1. Identify the requested quantities in each of the following situations involving exponential functions.
    1. What is the growth factor in the exponential function f(t) = 300(1.25)t.  What is the rate of growth as a percentage?  What is the initial amount?
    2. What is the change factor in the exponential function g(t) = 2(0.85)t.  What is the rate of decrease as a percentage?  What is the initial amount?
    3. What is the change factor in the exponential function h(t) = 6(0.5)2t+1.  What is the rate of decrease as a percentage?  What is the initial amount?  Warning: You must use rules of exponents first to put this in the proper form first.

    Solution.

  2. For each of the following situations, write down the corresponding exponential function relating A and t.
    1. You know that the amount, A, of  a radioactive sample decreases by 25% each week, and you begin with 5 grams, give an equation between A and t = the number of weeks after the initial weighing.
    2. You put $400 in the bank at 8% annual compound interest, give an equation between A = the amount in your account and t = the number of years, assuming you make no withdrawals.
    3. Babies grow rather quickly from a single cell into a many-celled embryo.  Assuming that A = the number of cells in a baby's body grows by a factor of 8 every minute, give an equation between A and t = the number of minutes after conception.

    Solution.

Banking, Interest, and Exponential Functions

  1. Use the correct compound interest formula to determine the following amounts.  Assume that you make no payments or withdrawals.
    1. You put $800 in the bank at 4% compounded weekly, determine the amount you will have in 10 years.
    2. You put $800 in the bank at 4% compounded continuously, determine the amount you will have in 10 years.
    3. You put $200 on a credit card that charges 18% compounded daily, determine the amount you will owe in 3 years.
    4. You take out a $15000 car loan at 8% compounded monthly, determine the amount you will owe in 5 years.
    5. You want to put some money into a savings account earning 6% compounded daily so that at the end of 4 years, you will have $3000 to take a post-graduation trip to Europe.  Determine how much money you will need to put in the account.  Hint: You will need to leave the initial amount, I, as an unknown in your equation, so that you can simplify and solve for it.

    Solution.

  2. Use the correct effective interest formula to determine the effective interest rate, reffective, in each situation.  Verify that using the effective interest rate compounded annually gives the same results as in the previous Exercise.
    1. Determine the effective annual rate corresponding to 4% compounded weekly.  Verify that $800 growing annually at this rate gives the same amount as before after 10 years.
    2. Determine the effective annual rate corresponding to 4% compounded continuously.  Verify that $800 growing annually at this rate gives the same amount as before after 10 years.
    3. Determine the effective annual rate corresponding to 18% compounded daily.  Verify that $200 growing annually at this rate gives the same amount as before after 3 years.
    4. Determine the effective annual rate corresponding to 8% compounded monthly.  Verify that $15000 growing annually at this rate gives the same amount as before after 5 years.

    Solution.

  3. You might think that the loan examples in the previous Exercise, were not very realistic, since we assumed that we would not make any payments on the loans.  A more realistic formula applies when you pay a fixed amount, p, each period:  

    Notice that this looks very similar to the previous formula, except we take a bit ( ) off the initial loan amount, I, and add it back at the end.  Note: Accountants have a nice, intuitive explanation of this formula in terms of the, so called, "time value of money".

    1. Assume that you put $500 on a credit card that charges 18% compounded monthly, and you make the minimum payment of $10 each month, use this formula to determine how much you would owe in 3 years.
    2. This formula can work to your advantage if you regularly put a fixed amount into an interest-bearing account.  Simply replace p by -p to derive the formula for your balance in a compound interest account if you put in p dollars per period.  
    3. Use your formula from part b) to determine how much money you will have in 45 years, if you start with $500 in the bank at 10% compounded weekly and you add $10 per week.

    Solution.

Go to Constructing Exponential Models .

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