Expert: Abe Mantell - 6/4/2011

Question
The volume of air in the lungs of a certain long distance runner is modelled by the eqauation
V=400sin(60pie t)+900
where V is the volume in cm3
t is time in minutes
1. what is the difference between the maximum and minimum volume of air in the runners lungs?
2. How many breaths per minute is the runner taking?
3. After one second id the runner breathing in or breathing out? Must explain your answer?
4. in each breath for how many seconds does the runner have at least 1200 cm3 of air in their lungs?
Please shoe me whole working so that I can find out how to show working?

Answer
Hello Rajvir,

1. V_minimum occurs when the sine function is a minimum, which is -1 (the sine function
.  goes from -1 to +1...V_maximum occurs when the sine function is a maximim (+1).
.  Thus, V_min=-400+900=500 cm^3, and V_max=+400+900=1300 cm^3...so the difference
.  is 1300-500=800 cm^3 (which is just 2 times the amplititude!)

2. Since t is in minutes, the period is (2pi)/(60pi)=1/30 of a minute...
.  which means that it takes 1/30 of a minute to complete one "breath cycle"
.  so, 30 breaths are taken per minute.

3. The period is 1/30 of a minute, which is 2 seconds (since there are 60 seconds/minute).
.  So, after 1 second it is halfway through a breath.  The breath starts with an inhale
.  (since the sine function starts out as an increasing function, so the volume of air
.  in the lungs starts out increasing)...halfway is at the time when the person has
.  just exhaled but not yet all of the air in the lungs...that will happen during the
.  next 1/2 second...so he (or she) is in the middle of an exhale!

4. A graph will be helpful...to see it, graph the function V=400sin(60pi t)+900
.  along with V=1200 to see where the volume is 1200 or more.  So we solve for where
.  400sin(60pi t)+900=1200 ==> 400sin(60pi t)=300 ==> sin(60pi t)=0.75...we get
.  t is about 0.0045 and 0.0122 (see attached graph).  Thus, per breath, there is
.  atleast 1200 cm^3 of air  for 0.0122-0.0045=0.0077 seconds.

OK?

Abe