I posted this question earlier incorrectly: I could still use some help figuring it out.
A student deposits $6,000 in a savings account with 6% continuously compounded interest. How many years must he wait until the balance has doubled?
12000 = 6000(e^(.06t) )
2 = e^(.06t)
take ln of both sides
ln2 = ln(e^(.06t))
ln2 = .06t (lne) , but lne = 1
.06t=ln2
t = ln2/.06
t = appr 11.55 years
Which symbol correctly compares these fractions?
13 7
___ __
20 10
__
No problem, I'd be happy to help you figure out the answer to your question!
To determine the number of years it takes for the balance to double, we can use the formula for compound interest:
A = P * e^(rt)
Where:
A = final amount
P = principal amount (initial deposit)
r = annual interest rate
t = time in years
e = Euler's number, approximately 2.71828
In this case, the principal amount (P) is $6,000, the annual interest rate (r) is 6%, and we need to find the time (t) it takes for the balance to double, which means the final amount (A) will be $12,000.
Plugging these values into the formula, we get:
$12,000 = $6,000 * e^(0.06t)
Now, we need to solve this equation to find the value of t. To isolate the variable, we can divide both sides of the equation by $6,000:
$12,000 / $6,000 = e^(0.06t)
2 = e^(0.06t)
To solve for t, we can take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(0.06t))
Using the property of logarithms, ln(e^(0.06t)) simplifies to 0.06t * ln(e). Since ln(e) is equal to 1, we have:
ln(2) = 0.06t
Now, divide both sides of the equation by 0.06:
ln(2) / 0.06 = t
Using a calculator, we can find the natural logarithm of 2 divided by 0.06 to get the value of t. The result will give us the number of years it takes for the balance to double.
So, to recap:
1. Calculate ln(2) / 0.06 using a calculator.
2. The result is the number of years it takes for the balance to double.