$1.$

A Ponzi scheme is a fraudulent investment operation in which returns to investors are paid from funds collected from new investors rather than from profit earned by the operator. The scheme takes its name from the notorious operation of Charles Ponzi in $1920.$ The case of Bernie Madoff is a more recent example.t Suppose the operator of a Ponzi scheme pays an initial return to investors of $$20,000.$ Each month, he must recruit enough new investors to increase the return by $3\%.$

$($a$)$ Find a formula that gives the return $R,$ in dollars, that the operator must pay after $t$ months.

$R(t)=$

$($b$)$ How much must the operator pay to investors at the end of $3$ years? $($Round your answer to two decimal places.$)$

$

$($c$)$ Assume that new investors pay $$2000$ to join the scheme. How many new investors must be recruited at the end of $3$ years in order to pay the existing investors? $($Enter a whole number of new investors.$)$

new investors

$2.$

A report tells us that in $2009,$ there were $870$ gray wolves in Idaho, but that the population declined by $19\%$ that year. For purposes of this problem, we assume that this $19\%$ annual rate of decrease continues.

$($a$)$ Find an exponential model that gives the wolf population $W$ as a function of the time $t$ in years since $2009.$

$w=$

$($b$)$ It is expected that the wolf population cannot recover if there are fewer than $30$ individuals. How long must this rate of decline continue for the wolf population to reach $30?($Round your answer to two decimal places.$)$

$$