QuestionIf 8000 dollars is invested in a bank account at an interest rate of 9 per cent per year, compounded continuously. How many years will it take for your balance to reach 20000 dollars?
Studdy Solution
STEP 1
1. The initial investment is 20000.
STEP 2
1. Use the formula for continuous compounding.
2. Substitute the known values into the formula.
3. Solve for the unknown variable (time in years).
STEP 3
Use the formula for continuous compounding, which is given by:
where:
- is the amount of money accumulated after years, including interest.
- is the principal amount (the initial amount of money).
- is the annual interest rate (in decimal).
- is the time in years.
- is the base of the natural logarithm.
STEP 4
Substitute the known values into the formula. We have:
-
-
-
Substituting these values, we get:
STEP 5
Solve for the unknown variable .
First, divide both sides by 8000 to isolate the exponential term:
Simplify the fraction:
Take the natural logarithm of both sides to solve for :
Using the property of logarithms that , we have:
Now, solve for by dividing both sides by 0.09:
Calculate the value:
Therefore, it will take approximately years for the balance to reach $20000.
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