Keith makes an investment of $$4000$ into an account which pays an interest rate of $5\mathrm{.}45\%$ per year, compounded quarterly.

a$.$ One form of the compound interest formula is $A=P(1+i{)}^n,$ where $A$ is the amount of money at the end of the investment; $P$ is the principal amount deposited; $i$ is the interest rate per compounding period; and $n$ is the number of compounding periods.

Use the above formula to algebraically determine the number of compounding periods it would take for the investment to double. $($Hint: Review page $362$ in the textbook to help you set up this equation and then solve using the logarithmic laws.$)$ State your value for $n$ to one decimal place. $(2$ marks$)$

b$.$ Explain why the correct solution for the number of compounding periods is $52$ and not $51.(0\mathrm{.}5$ mark$)$

c$.$ Another form of the compound interest formula is $A=P(1+\frac{r}{n}{)}^{nt},$ where $A$ is the amount of money at the end of the investment; $P$ is the principal amount deposited; $r$ is the interest rate per year; $n$ is the number of compounding periods a year; and $t$ is the total time in years.

Using this second formula graphically determine the time in years it would take for the investment to double. Show your work by completing the statements below for your graphing calculator steps. Explain how this answer verifies your results from part a$.(1\mathrm{.}5$ marks$)$

$y_1=$

$y_2=$

$t=$