Your lungs hold about 4.5 liters of air after you inhale and about 3.5 liters of air after you exhale. If you take 6 breaths every minute, then a model for the amount of air in your lungs at t minutes is

Step-by-step explanation:

Function Modeling

Since the air in the lungs goes periodically from a minimum value to a maximum value, we can simulate its behavior as a sinusoid. Selecting a sine or a cosine will depend on the initial condition we'll assume since the question doesn't provide one. We'll set the initial state when the lungs are at maximum air content. The sinusoid that starts from maximum is the cosine, so our model is

Where Ao is the amplitude of the oscillation, w is the angular frequency and M is the midline or y-displacement of the wave.

The values of A run from a min of 3.5 to a max of 4.5. That gives us twice the amplitude, thus

The vertical displacement or midline can be found as the shift from the center value:

We also know the cycle repeats 6 times per minute. If the time is expressed in minutes, then the frequency is f=6

Knowing that

Then

The model is

where t is expressed in minutes

amount of oatypop that contains the prize = 40%

amount   of oatypop which doesn't contain the prize = (100-40)=60%

probability that hannah will win a prize

    = s +f s +f f s+ff   f s+ff f f s+ infinity.where f= failure,s= success

=   or 60%

probability that hannah only has to buy 3 or less boxes before getting a prize= (s) and ( f s) and (ff s) and (f f f s)

        = = 87.04 %  

step-by-step explanation: