Suppose that $17,490 is invested at an interest rate of 6.4% per year, compounded continuously.
a) Find the exponential function that describes the amount in the account after time t, in years.
b) What is the balance after 1 year? 2 years? 5 years? 10 years?
c) What is the doubling time?
a) The exponential growth function is P(t)=.
(Type exponential notation with positive exponents. Do not simplify. Use integers or decimals for any numbers in the equation.)
b) The balance after 1 year is $
(Simplify your answers. Round to two decimal places as needed.)
The balance after 2 years is $
(Simplify your answers. Round to two decimal places as needed.)
The balance after 5 years is $
(Simplify your answers. Round to two decimal places as needed.)
The balance after 10 years is $
Answer (2)
The exponential function describing the amount in the account is P ( t ) = 17490 e 0.064 t . The balances after 1, 2, 5, and 10 years are approximately $18643.24, $19863.41, $23999.41, and $32665.14, respectively. The doubling time is approximately 10.83 years.
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The exponential growth function is P ( t ) = 17490 e 0.064 t .
The balance after 1 year is approximately $18643.24, after 2 years is approximately $19863.41, after 5 years is approximately $23999.41, and after 10 years is approximately $32665.14.
To find the doubling time, solve for t in the equation 2 = e 0.064 t , which gives t = 0.064 l n ( 2 ) ≈ 10.83 years.
The doubling time is approximately 10.83 years.
Explanation
Problem Analysis We are given an initial investment of $17,490 at an interest rate of 6.4% per year, compounded continuously. We need to find the exponential function that describes the amount in the account after time t, the balance after 1, 2, 5, and 10 years, and the doubling time.
Formula for Continuous Compounding The formula for continuous compounding is given by: P ( t ) = P 0 e r t where:
P ( t ) is the amount in the account after time t,
P 0 is the initial investment,
r is the interest rate (as a decimal),
t is the time in years.
Finding the Exponential Function a) To find the exponential function, we substitute the given values into the formula: P 0 = 17490 r = 6.4% = 0.064 So, the exponential function is: P ( t ) = 17490 e 0.064 t
Calculating the Balance After Different Years b) Now, we need to find the balance after 1, 2, 5, and 10 years. We will substitute t = 1, 2, 5, and 10 into the exponential function we found in part a. For t = 1 year: P ( 1 ) = 17490 e 0.064 ( 1 ) = 17490 e 0.064 ≈ 18643.24 For t = 2 years: P ( 2 ) = 17490 e 0.064 ( 2 ) = 17490 e 0.128 ≈ 19863.41 For t = 5 years: P ( 5 ) = 17490 e 0.064 ( 5 ) = 17490 e 0.32 ≈ 23999.41 For t = 10 years: P ( 10 ) = 17490 e 0.064 ( 10 ) = 17490 e 0.64 ≈ 32665.14
Finding the Doubling Time c) To find the doubling time, we need to find the time t when the amount in the account is double the initial investment. So, we set P ( t ) = 2 P 0 = 2 ( 17490 ) = 34980 .
34980 = 17490 e 0.064 t Divide both sides by 17490: 2 = e 0.064 t Take the natural logarithm of both sides: l n ( 2 ) = 0.064 t Solve for t: t = 0.064 l n ( 2 ) ≈ 10.83
Final Answer Therefore: a) The exponential growth function is P ( t ) = 17490 e 0.064 t .
b) The balance after 1 year is approximately $18643.24, after 2 years is approximately $19863.41, after 5 years is approximately $23999.41, and after 10 years is approximately $32665.14. c) The doubling time is approximately 10.83 years.
Examples
Understanding continuous compounding is useful in various financial scenarios. For instance, when planning for retirement, you can use this concept to project the growth of your investment portfolio over time. By knowing the interest rate and the time horizon, you can estimate how long it will take for your investments to reach your financial goals. This knowledge helps in making informed decisions about savings, investments, and retirement planning, ensuring a financially secure future.
