I am sure this is the last time i am bothering you steve for today! you are very helpful.
1) I have to find the population density i which r is 0.2 apart and distance is 1.2km so there are total 7 values from 0 to 1.2 , i am taking right and left end-point approximations and taking average of them, still not getting right answer?
2) Breathing cycle takes about 5 seconds.Maximum rate of flow is 0.5/L. The function often use to model the rate of air flow is 0.5sin(2pi*t/5). And volume of air inhaled in the lungs at time is given by 5/4*pi [1-cos(2*pi*t/5)]L. Find average volume of inhaled air in the lungs in one respiratory cycle
the density will be the (average value of the population) divided by the area
just as we had to integrate the density times area to get total population
.
Remember that the average value of f(x) in the interval [a,b] is
∫[a,b] f(x) dx
------------------
b-a
1) To find the population density with a distance of 1.2 km and intervals of 0.2 km, you need to calculate the average value of the population density over those intervals.
First, determine the number of intervals by dividing the total distance (1.2 km) by the interval size (0.2 km). In this case, 1.2 km divided by 0.2 km gives you a total of 6 intervals.
Now, you need to calculate the population density at each endpoint of the intervals. The right endpoints would be 0.2, 0.4, 0.6, 0.8, 1.0, and 1.2 km. Similarly, the left endpoints would be 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 km.
Calculate the population density at each endpoint using the given information or data. Sum up all the population densities at the right endpoints and divide it by the total number of intervals. Then, do the same for the left endpoints. Finally, find the average of the two values obtained. This average will give you the average population density over the given distance.
If you are not getting the right answer, you may want to recheck your calculations or clarify the specific data or formula used to calculate the population density.
2) To find the average volume of inhaled air in one respiratory cycle, you need to integrate the given function over one complete cycle and divide it by the time period of the cycle.
The function representing the volume of air inhaled in the lungs at any given time, based on the given information, is 5/4 * π * [1 - cos(2 * π * t/5)] L, where t represents time in seconds.
To find the average volume, you need to integrate this function over one respiratory cycle, which takes about 5 seconds. The limits of integration will be from 0 to 5.
Average Volume = (1/5) * ∫[0, 5] [5/4 * π * [1 - cos(2 * π * t/5)] dt.
Simplify and solve the integral to find the average volume.