Final answer:
To model a bank account with $20,000 and earning 3% interest compounded annually, the equation is $20,000 = $20,000(1 + 0.03/1)^(1*0). The interest does not compound over any number of years, so the balance will remain $20,000.
Step-by-step explanation:
To model the bank account with $20,000 and earning 3% interest compounded annually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where A is the final amount, P is the principal amount (initial deposit), r is the interest rate (in decimal form), n is the number of times interest is compounded per year, and t is the number of years.
Plugging in the values from the question, we have:
- A = final amount = $20,000
- P = principal amount = $20,000
- r = interest rate = 3% = 0.03
- n = number of times compounded per year = 1 (annually)
- t = number of years = unknown, we'll solve for it
Substituting these values into the formula, we have:
$20,000 = $20,000(1 + 0.03/1)^(1*t)
Now we can solve for t:
Dividing both sides by $20,000:
1 = (1 + 0.03/1)^(1*t)
Since the base for the exponential is 1 + 0.03/1 = 1.03, we have:
1 = 1.03^(1*t)
Take the natural logarithm (ln) of both sides to isolate the exponent:
ln(1) = ln(1.03^(1*t))
0 = 1*t * ln(1.03)
ln(1) is 0, so we're left with:
0 = t * ln(1.03)
Dividing both sides by ln(1.03):
0 / ln(1.03) = t
t ≈ 0
Therefore, the equation that models the bank account is: $20,000 = $20,000(1 + 0.03/1)^(1*0)
The interest is not compounding over any number of years, which means the balance will remain $20,000.