1) For a person at rest, the velocity v (in liters per second) of air flow during a respiratory cycle is
V = 0.85 sin pi(t)/ 3
where t is the time in seconds. (Inhalation occurs when v > 0 and exhalation occurs when v < 0)
A) Find the time for one full respiratory cycle
B) Find the number of cycles per minute.
Please explain. I do not understand!! Thank You
Well, breathing can be a bit of a tricky business, but let me clown around and explain it to you in a fun way!
A) To find the time for one full respiratory cycle, we need to find when v goes from positive to negative and back to positive again. In other words, we need to find the two points where v = 0.
Now, the equation for v given is V = 0.85 sin(pi(t)/3). So, when is sin(pi(t)/3) equal to zero? Think about it - sin(pi/2) equals zero, right? That means pi(t)/3 must equal pi/2, or t/3 = 2.
Solving for t, we get t = 6. So, it takes 6 seconds for one full respiratory cycle.
B) To find the number of cycles per minute, we need to convert 6 seconds to minutes. There are 60 seconds in a minute, so 6 seconds is equal to 1/10th of a minute.
Since we know it takes 1/10th of a minute for one respiratory cycle, we can now calculate the number of cycles per minute by taking the reciprocal - that's a fancy way of saying we flip the fraction.
So, 1/ (1/10) = 10 cycles per minute!
I hope that explanation brings a breath of fresh air to your understanding!
To find the time for one full respiratory cycle, we need to find the period of the function V(t). The period of a sine function is given by the formula T = 2π/b, where b is the coefficient of t in the sine function.
In this case, the coefficient of t is π/3. So, the period T = 2π / (π/3) = 6 seconds. Therefore, one full respiratory cycle takes 6 seconds.
To find the number of cycles per minute, we need to find the frequency of the function V(t). The frequency of a function is the reciprocal of its period. So, the frequency f = 1 / T = 1 / 6 cycles per second.
To convert cycles per second into cycles per minute, we multiply by 60 (since there are 60 seconds in a minute). Therefore, the number of cycles per minute is 1/6 * 60 = 10 cycles per minute.
So, the answers are:
A) One full respiratory cycle takes 6 seconds.
B) There are 10 cycles per minute.
A) To find the time for one full respiratory cycle, we need to determine the time it takes for the velocity to complete one full period. In this case, one full period corresponds to when the sine function completes one full oscillation.
The general formula for the period of a sine function is given by:
T = 2π/ω
Where T is the period, and ω is the angular frequency. In our case, the angular frequency is given by π/3. Plugging the value of ω into the formula, we have:
T = 2π / (π/3) = 6
So, the time for one full respiratory cycle is 6 seconds.
B) To find the number of cycles per minute, we need to convert the time for one cycle to minutes and then take the reciprocal.
There are 60 seconds in a minute, so we can convert our result from part A by dividing it by 60:
6 seconds / 60 = 0.1 minutes
Finally, taking the reciprocal, we find the number of cycles per minute:
1 / 0.1 = 10 cycles per minute
Therefore, there are 10 cycles per minute during respiration.
Hope that helps! Let me know if you have any further questions.
The sin function is periodic ever 2PI radians.
a)2PI=PI/3 T
or T for one cycle=6 seconds
b) how many 6 seconds are in one minute?