$2$ Problem $2$

On average in the United States, credit card interest has an Annual Percentage Rate $($APR$)$ of $22\mathrm{.}76\%$ according to the Federal Reserve as of May $2024,$ and compounds daily. $($Note that if you're just opening a credit account or have only had one for a year or so$,$ your rate will most likely be higher$).$

a$)$ Say you spend $$1200$ in December $2023$ with your credit card, meaning you owe this much to the credit card company as of January $1$st$,2024.$ Show $($or describe in words$)$ that your total debt as a function of time $($in months$)$ is given by the function

$D(t)=1200(1+\frac{0\mathrm{.}2276}{365}{)}^{\frac{365**t}{12}}$

$1$

b$)$ Draw a sketch of this function $($including labelled axes and any important points$)$ with an appropriate choice of domain. Given your choice of domain, what is the range?

c$)$ How much additional money do you owe the credit card company after $1$ month? $3$ months? $1$ year?

d$)$ Given that $lo{g}_b(x)$ is the inverse function of $b^x,$ algebraically determine about how many years it takes for you to double your initial debt?

e$)$ Most credit card companies only require you to pay a small piece of your debt at the end of each month; for example Discover only requires a $$35$ payment to avoid late fees. Keeping in mind that our function is the compounding of only $1$ month of spending, use our model to justify why you should pay your balance off in full at the end of each month, and not simply pay $$35.$