Question

the formula ( a = pe^{rt} ) describes the accumulated value, ( a ), of a sum of money, ( p ), the principal, after ( t ) years at annual percentage rate ( r ) (in decimal form) compounded continuously. complete the table for a savings account subject to continuous compounding.

amount invested	annual interest rate	accumulated amount	time ( t ) in years

(do not round until the final answer. then round to one decimal place as needed.)

Explanation:

Step1: Identify given values

We know the formula for continuous compounding is \( A = Pe^{rt} \). Here, \( P=\$9500 \), \( r = 9\%=0.09 \), and \( A = 2P \) (since we want to double the amount invested). So \( A = 2\times9500 = 19000 \).

Step2: Substitute into the formula

Substitute \( A = 19000 \), \( P = 9500 \), and \( r = 0.09 \) into \( A = Pe^{rt} \):

$$ 19000=9500e^{0.09t} $$

Step3: Solve for \( t \)

First, divide both sides by 9500:

$$ \frac{19000}{9500}=e^{0.09t} $$

Simplify the left side: \( 2 = e^{0.09t} \)

Then, take the natural logarithm of both sides:

$$ \ln(2)=\ln(e^{0.09t}) $$

Using the property \( \ln(e^x)=x \), we get:

$$ \ln(2)=0.09t $$

Now, solve for \( t \) by dividing both sides by 0.09:

$$ t=\frac{\ln(2)}{0.09} $$

Calculate \( \ln(2)\approx0.6931 \), so:

$$ t=\frac{0.6931}{0.09}\approx7.7 $$

Answer:

\( 7.7 \)