A savings account earns 4 percent interest per year, compounded continuously. If 2200 dollars is deposited, how much is in the account after seven years with no withdrawals? Round to the nearest cent.

Step-by-step solution

1. 1

   Use the continuous-compounding formula. Continuous compounding is the limit of compounding more and more often, and it has its own closed form:

   $A=Pe^{rt}.$

2. 2

   List the inputs with the rate as a decimal.

   $P=2200,\qquad r=4\%=0.04,\qquad t=7\ \text{years}.$

3. 3

   Compute the exponent.

   $rt=0.04\times 7=0.28.$

4. 4

   Evaluate the exponential at full precision.

   $e^{0.28}=1.32312981\ldots$

   Rounding here to $1.323$ would cost about twenty-nine cents in the final answer, which is why the problem forbids intermediate rounding.

5. 5

   Multiply and round.

   $A=2200\times 1.32312981=2910.88559\ \Longrightarrow\ \$2910.89.$

6. 6

   Check the size against simple interest. Simple interest would give 2200(1.28)=\2816$, and continuous compounding must exceed it. The extra$$94.89$is the interest earned on interest over the seven years — a reasonable amount at$4%$.