Calculus question with given formula (ten points)?

Question

The fraction of toy boats that sink in less than t days is approximately:

f(t) = 1 - e^-0.02t

("F" of "t" equals 1 minus e to the power of negative 0.02t)

Question: What fraction can float for at least 12 days?

I don't want the answer, I want to know HOW to do the question. This whole "at least" 12 days is throwing me off. Any help is greatly appreciated, thanks :)

Answer 1

Find out what fraction sink by day 12. Subtract that from one to find the fraction that float for twelve days or more :)

Answer 2

Plug in t = 12



f(12) = 1 - e^ -0.02(12)



f(12) = 0.213 <--- This is the fraction that has sunk in 12 days.





We want to know the fraction that HAS NOT sunk.



1 - 0.213 = 0.787 <--- answer







If you did not understand why I subtracted 0.213 from 1, then consider the following:



2/3 of the boats have sunk, that means



1/3 of the boats have not sunk.





You get 1/3 by doing



1 - 2/3



= 3/3 - 2/3



= 1/3

Answer 3

It would really help to know the units of t. But as t increases what happens to f(t). I am going to assume t is in days.



f(12) = 1 - e^-(.02t)

f(12) = 1 - e^-(0.02*12)

f(12) = 1 - e^-(0.24)

f(12) = 0.2133 This is the fraction that sinks.

1 - 0.2133 should be the number that floats which is about 0.8



What happens when you decide you want to go for 24 days?

f(24) = 1 - e^(-0.02*24)

f(24) = 0.38

1 - 0.38 = the number that should float which is smaller. (about 0.6)



So 12 days is the maximum. After that more and more of them sink.

Answer 4

I think the entire negation aspect of this questioning is confusing you (and me)!



Notice that t is a continuous variable, not discrete integer days.



Also forget the "approximately". We are modelling a situation and the model is an approximation, but the function is perfect for the model!



Fraction that "sink in less than" t days + Fraction that did not "sink in less than" t days = all = 1



So

Fraction that did not "sink in less than" t days = e^-0.02t



But these are the ones that "float for at least" t days.

"float for at least" means "did not sink before"



Notice the less than t days" means
and "at least t days" means >= t.

Answer 5

okay, so what you do is plug in 12 for t and that will tell you how many sink in less than 12 days. You will get .2133721 or 21.337% will sink in less than 12 days. Therefore, about 78.663% can float for at least 12 days.

Answer 6

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