During a regular respiratory cycle, the volume of air in liters in the human lungs can be described by the function V(t) = 0.173t +0.15t^2 - 0.035t^3 where t is the time in seconds. Estimate the time in seconds from the beginning of this respiratory cycle for the lungs to fill to their maximum volume of air. a. about 3.4 seconds b. about 4.4 seconds c. about 5.4 seconds d. about 10.4 seconds

Step-by-step explanation:

Given the volume of air in liters in the human lungs described by the function V(t) = 0.173t +0.15t^2 - 0.035t^3 where t is the time in seconds,

The beginning of this respiratory cycle for the lungs to fill to their maximum volume of air occurs when dV/dt = 0

dV/dt = 0.173 + 0.3t - 3(0.035)t²

dV/dt = 0.173 + 0.3t - 0.105t²

Since dV/dt = 0

0.173 + 0.3t - 0.105t² = 0

multiply through by -1

-0.173 - 0.3t + 0.105t² = 0

0.105t² -0.3t-0.173 = 0

Factorize;

t = 0.3±√0.3²+4(0.105(0.173))/2(0.105)

t = 0.3±√0.09+0.07266/0.21

t =  0.3±√0.16266/0.21

t = 0.30±0.4033/0.21

t = 0.7033/0.21

t = 3.349secs

t ≈ 3.4secs

Hence it will take about 3.4secs for the lungs to fill to their maximum volume of air

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