Respuesta :
Answer:
The average volume of air in the lungs during one cycle is 0.53176 liters.
Step-by-step explanation:
Given : The volume V, in liters, of air in the lungs during a five-second respiratory cycle is approximated by the model [tex]V=0.1729t+0.1522t^2- 0.0374t^3[/tex], where t is the time in seconds.
To find : The average volume of air in the lungs during one cycle ?
Solution :
The volume V of air in the lungs during a five-second respiratory cycle is [tex]V=0.1729t+0.1522t^2- 0.0374t^3[/tex]
Then the average volume of air in the lungs during one cycle [0,5] is
[tex]V_a=\frac{1}{b-a}\int\limits^a_b {V(t)} \, dt[/tex]
[tex]V_a=\frac{1}{5-0}\int\limits^0_5 {0.1729t+0.1522t^2- 0.0374t^3} \, dt[/tex]
[tex]V_a=\frac{1}{5}[\frac{0.1729t^2}{2}+\frac{0.1522t^3}{3}- \frac{0.0374t^4}{4}}]^5_0[/tex]
[tex]V_a=\frac{1}{5}[0.08645t^2+0.05073t^3- 0.00935t^4]^5_0[/tex]
[tex]V_a=\frac{1}{5}[0.08645(5)^2+0.05073(5)^3- 0.00935(5)^4-0.08645(0)^2+0.05073(0)^3- 0.00935(0)^4][/tex]
[tex]V_a=\frac{1}{5}[2.1613+6.3413-5.8438][/tex]
[tex]V_a=\frac{1}{5}[2.6588][/tex]
[tex]V_a=0.53176\ l[/tex]
The average volume of air in the lungs during one cycle is 0.53176 liters.
