1. Anita is trying to be more deliberate about saving money, so she opens a high-yield money-market account at her local bank. As an exercises in self-discipline she commits to depositing three dollars into the account every morning. The account offers a 2% annual interest rate compounded daily. Model her account balance as if her deposits were a continuous money flow into her account and as if the bank were compounding her interest continually, and calculate the projected balance of her account after 4 years. How much less total money is this than if she had instead deposited her entire annual contribution, $1095, at the beginning of each year?
2. Billiam’s employer offers retirement benefits. In total, his employer contributes $12,000 per year into a 401K retirement account that’s tied to the stock market and appreciates by 7% annually on average. Billiam’s employer and the financial institution have an agreement that the contributions be made on a continual basis. What will the accumulated future value of Billam’s retirement account be after 10 years? How long before his account balance crosses the million dollar threshold?
3. Clyde and Clarice just had a baby. They named him Carl. They’d like to establish a trust fund for li’l Carl that will appreciate to a value of one million dollars by his eighteenth birthday. Their financial adviser points them to a high-yield investment that offers a 6% annual return on average. What lump-sum would they have to invest now to appreciate as planned? Unable to afford this amount now, how much would they have to contribute to the investment annually, deposited continually, for it to appreciate as planned?
4. Wait times for a 911 call can be modelled with an exponential distribution: if the continuous random variable \(X\) represents a given wait time in seconds, then the probability density function (PDF) \(f(x) = 0.21 \mathrm{e}^{-0.21 t}\) models the probability that a person calling 911 has to wait that many seconds before an operator answers. According to this PDF, upon calling 911, what is the probability that an operator will answer in the first five seconds? What is the probability that an operator will take longer than ten seconds to answer?