Question

If $500 were to compound
continuously at a yearly interest
rate of 7%, what would the total
amount be after 10 years?
$[?]
Round your answer to the nearest hundredth.

Answer 1

Answer:

Total amount: $1,006.88

Step-by-step explanation:

Given the principal amount of $500 that is compounded continuously for 10 years at an annual interest rate of 7%:

We can use the following Continuous Compound Interest Formula to determine the future value of the total amount of investment:

$\displaystyle\mathsf{\Bigg\ Continuous\:Compound\:Interest:\quad\ A\:=\:Pe^{(r\times\\ t)}}$

where:

A = The future value of the total amount in the account at the end of "t" number of years

P = Present value of the principal amount invested = $500

e = constant (base of the exponential function) ≈ 2.71828

r = Annual interest rate = 7% or 0.07

t = time (in years)  = 10 years

Solution:

Substitute the given values into the  Continuous Compound Interest Formula:

$\displaystyle\mathsf{\Bigg\ Continuous\:Compound\:Interest:\quad\ A\:=\:Pe^{(r\times\\ t)}}$

$\displaystyle\mathsf{A\:=\: 500\:\times\ 2.71828 ^{(0.07\times\\ 10)}}$

$\displaystyle\mathsf{A\:=\: 500\:\times\ [2.71828 ^{(0.70)}]}$

$\displaystyle\mathsf{A\:=\: 500\:\times\ 2.013752707}$

$\displaystyle\mathsf{A\:=\ 1,006.88}}$  

Therefore, the total amount accumulated after continuously compounding the principal investment for 10 years is $1,006.88. This includes the principal amount invested, $500, plus the interest accrued of $506.88.