Answer:
Explanation:
(a) The formula to calculate the amount of money in the account after t years with an initial deposit of P and an annual interest rate r compounded annually is:
A = P(1 + r)^t
In this case, P = $500, r = 0.015 (1.5% expressed as decimal), and the interest is compounded annually, so the equation is:
A = 500(1 + 0.015)^t
Simplifying this gives:
A = 500(1.015)^t
Therefore, the equation that represents the amount of money in the savings account as a function of time in years is A = 500(1.015)^t.
(b) To find the amount of time it takes for the account balance to reach $800, we can set the equation A = 500(1.015)^t equal to 800 and solve for t:
500(1.015)^t = 800
Dividing both sides by 500:
(1.015)^t = 1.6
Taking the natural logarithm of both sides:
ln(1.015)^t = ln(1.6)
Using the power rule of logarithms:
t ln(1.015) = ln(1.6)
Dividing both sides by ln(1.015):
t = ln(1.6) / ln(1.015)
Using a calculator:
t ≈ 24.7
Therefore, it takes approximately 24.7 years for the account balance to reach $800.