Exponential Models: The magistrate states that the penalty is $1 million, to be paid on July 01 2016 and the fine increases by $10 million each day thereafter. The company's legal counsel insists that the penalty is unfair and arbitrary and hence they are not legally bound to pay the fine until an appeal case is heard.

Penalty A: $1 million dollars to be paid on July 2016 and the fine increases by $10 million each day thereafter i.e. the original penalty.

Penalty B: An initial amount of 1 cent, beginning on July 01 and doubling thereafter, with the fine payable to be the amount generated after 40 days of doubling.

1. The company legal counsel is delighted and without recourse to a consulting mathematician, readily take the offer of Penalty B, thinking that a penalty of 1 cent cannot double sufficiently to be anywhere near Penalty A.

Which penalty will be larger after the 40 days? Were the company counsel correct in taking the option of Penalty B?

Express as a formula, the amount Penalty A as a function of time t , where t is the number of days after July 01 2016

Express as a formula, the amount Penalty B as a function of time t , where t is the number of days after July 01 2016.