If $8000 is invested in an account for which interest is compounded continuously, find the amount of the investment at the end of 12 years for the following interest rates.
a) 2%
b) 3%
c) 4.5%
d) 7%
a)
amount = 8000 e^(12(.02))
= 8500 e^.24
= $ 10805.62
repeat procedure for the others
To find the amount of the investment at the end of 12 years with continuously compounded interest, we can use the formula:
A = P * e^(rt)
Where:
A = the amount of the investment at the end of 12 years
P = the principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = the interest rate (as a decimal)
t = the time period in years
Let's plug in the given values for each interest rate:
a) For an interest rate of 2% (0.02 as a decimal):
A = 8000 * e^(0.02 * 12)
b) For an interest rate of 3% (0.03 as a decimal):
A = 8000 * e^(0.03 * 12)
c) For an interest rate of 4.5% (0.045 as a decimal):
A = 8000 * e^(0.045 * 12)
d) For an interest rate of 7% (0.07 as a decimal):
A = 8000 * e^(0.07 * 12)
Now, let's calculate each value step by step.
a) For an interest rate of 2%:
A = 8000 * e^(0.02 * 12)
= 8000 * e^(0.24)
≈ 8000 * 1.27124
≈ 10,169.92
b) For an interest rate of 3%:
A = 8000 * e^(0.03 * 12)
= 8000 * e^(0.36)
≈ 8000 * 1.43496
≈ 11,479.71
c) For an interest rate of 4.5%:
A = 8000 * e^(0.045 * 12)
= 8000 * e^(0.54)
≈ 8000 * 1.71616
≈ 13,728.21
d) For an interest rate of 7%:
A = 8000 * e^(0.07 * 12)
= 8000 * e^(0.84)
≈ 8000 * 2.32012
≈ 18,560.96
Therefore, the amount of the investment at the end of 12 years for each interest rate is:
a) $10,169.92
b) $11,479.71
c) $13,728.21
d) $18,560.96
To find the amount of the investment at the end of 12 years for each interest rate, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the amount of the investment at the end of the specified time period
P = the principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = the annual interest rate (in decimal form)
t = the time period (in years)
Let's calculate the amount for each interest rate:
a) For an interest rate of 2% (0.02 in decimal form):
A = 8000 * e^(0.02 * 12)
To calculate this, we need to use the value of Euler's number (e) raised to the power of (0.02 * 12). Once we have this value, we can multiply it by the principal amount (8000) to find the final amount of the investment.
b) For an interest rate of 3% (0.03 in decimal form):
A = 8000 * e^(0.03 * 12)
c) For an interest rate of 4.5% (0.045 in decimal form):
A = 8000 * e^(0.045 * 12)
d) For an interest rate of 7% (0.07 in decimal form):
A = 8000 * e^(0.07 * 12)
Now, let's evaluate each of these expressions to find the final amount of the investment at the end of 12 years for each interest rate.