If $3500 is invested at an interest rate of 6.25% per year, compounded continuously, find the value of the investment after the given number of years.
a) 3 years
b) 6 years
c) 9 years
To find the value of the investment after a certain number of years when it is compounded continuously, we can use the formula:
\[A = P e^{rt}\]
Where:
A = the final amount
P = the principal or initial investment
r = the interest rate
t = the number of years
e = Euler's number, approximately 2.71828
a) To find the value after 3 years:
P = $3500
r = 6.25% = 0.0625
t = 3
Plugging the values into the formula:
\[A = 3500 \times e^{0.0625 \times 3}\]
Calculating:
\[A = 3500 \times e^{0.1875}\]
\[A \approx 3500 \times 1.2067\]
\[A \approx $4223.35\]
Therefore, the value of the investment after 3 years is approximately $4223.35.
b) To find the value after 6 years:
P = $3500
r = 6.25% = 0.0625
t = 6
Plugging the values into the formula:
\[A = 3500 \times e^{0.0625 \times 6}\]
Calculating:
\[A = 3500 \times e^{0.375}\]
\[A \approx 3500 \times 1.4559\]
\[A \approx $5089.58\]
Therefore, the value of the investment after 6 years is approximately $5089.58.
c) To find the value after 9 years:
P = $3500
r = 6.25% = 0.0625
t = 9
Plugging the values into the formula:
\[A = 3500 \times e^{0.0625 \times 9}\]
Calculating:
\[A = 3500 \times e^{0.5625}\]
\[A \approx 3500 \times 1.7553\]
\[A \approx $6139.50\]
Therefore, the value of the investment after 9 years is approximately $6139.50.
To find the value of the investment after a certain number of years, we can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = the final amount
P = the principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = the interest rate (expressed as a decimal)
t = the time period in years
Let's calculate the value of the investment for each given time period:
a) 3 years:
A = 3500 * e^(0.0625 * 3)
First, calculate 0.0625 * 3 = 0.1875
Then, calculate e^(0.1875) using the approximate value of e
The value of the investment after 3 years is approximately 3500 * e^(0.1875).
b) 6 years:
A = 3500 * e^(0.0625 * 6)
First, calculate 0.0625 * 6 = 0.375
Then, calculate e^(0.375)
The value of the investment after 6 years is approximately 3500 * e^(0.375).
c) 9 years:
A = 3500 * e^(0.0625 * 9)
First, calculate 0.0625 * 9 = 0.5625
Then, calculate e^(0.5625)
The value of the investment after 9 years is approximately 3500 * e^(0.5625).
You can use a scientific calculator or an online calculator to find the approximate values of e^(0.1875), e^(0.375), and e^(0.5625) and then multiply them with $3500 to get the final amounts for each time period.
a)
amount = 3500 e^(3(.0625))
= .....
same for the others.