Question 1096702: Exponential Models: In a civil case a large company has a judgement made against it.
The magistrate states that the penalty is $1 million, to be paid on July 01 2017 and then fine
increases by $10 million each day thereafter. The companys 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. The magistrate is very canny and in response to this
they decide to set aside their intial ruling and instead makes the following offer of a choice
of penalties, with the chosen penalty to be legally binding once agreed to:
Penalty A: $1 million dollars to be paid on July 1 , 2017 and the ne increases by $10 million
each day thereafter i.e. the original penalty
Penalty B: An initial amount of 1 cent, beginning on July 1, 2017 and doubling thereafter, with the ne payable to be the amount generated after 40 days of doubling
The companylegal counsel is delighted andwithout recourse toaconsulting mathematician,
readily take the o⁄er of Penalty B, thinking that a penalty of 1 cent cannot double su¢ ciently
to be anywhere near Penalty A.
1. (a) Which penalty will be larger after the 40 days? Were the company counsel correct
in taking the option of Penalty B?
(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 2017.
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Looking at 40 days (which isn't after 40 days but is the boundary).
Penalty A is 1+(10)(40)=$401 million. The formula is Cost=$1+$10*t, where the dollars are in millions, and t is the number of days after 1 July ANSWER.
For penalty B, look at it in cents first.
This is the sum (sigma) of 2^t, with t taken from 0 to 40.
The sum of powers of n: are the highest power (6), add 1 to that power (7) and subtract 1 from the result.
{The sum of 1+2+4+8+16+32+64 is 127). Here, we have an initial amount of 1 cent or 2^0.
Day 1 is 1 or 2^0, on day 2 the sum is 3 cents (2^2)-1, day 3, 4+2+1 or 7 cents (2^3-1); day 4, 15 cents (8+4+2+1) or 2^4-1.
On day 40, it will be 2^40-1 cents.
That is 1.1 x 10 ^12 cents. While getting the exact number of cents is more difficult, in this instance we aren't asked for that but the relative costs.
This is $1.1 x 10^10, or $11 billion.
A formula in dollars is (1/100)(2^t)-1, where t= days, but this uses 1 July and not the days after which is what the question asks for. So, for t days after the first, it is 2^(t+1)-2 in cents (2+4+8+16+32+64+...2^40)
Penalty A is a far wiser choice. The company made the WRONG choice. ANSWER
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