see image below When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragmthrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction ofthe trachea, making a narrower channel for the expelled air to flow through. For a given amount of airto escape in a fixed time, it must move faster through the narrower channel than the wider one. Thegreater the velocity of the airstream, the greater the force on the foreign object. X rays show that theradius of the circular tracheal tube contracts to about two-thirds of its nOTmal radius during a cough.According to a mathematical model of coughing, the velocity v, of the airstream is related to the radiusT, of the trachea by the following equation:v (r) = k (ro – r)r?where k and To are pasitive constants.Sketch the graph of the function v (r) above, on the interval (0, ro), by following the steps below:(a) Find the values of v (r) for r = 0 andr =To-12](b) Find the intervals of increase or decrease. 19](c) Find the local maximum value. [2(d) Find the intervals of concavity and the inflection point.[13||(e) Use the information of parts (a)-(d) to sketch the graph (on axes v(r) vs r). 5|

a. The function vr=kr0-rr2v We calculate the values as v0=kr0-0·02v0=0 and vr0=kr0-r0·r02vr0=0 So…

thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity v, of the airstream is related to the radius r, of the trachea by the following equation: v (r) = k (ro - r)r2 where k and ro are positive constants. Sketch the graph of the function v (r) above, on the interval (0, ro], by following the steps below: (a) Find the values of v (r) for r =0 and r = ro.2| (b) Find the intervals of increase or decrease. 19| (c) Find the local maximum value. [2 (d) Find the intervals of concavity and the inflection point.[13| (e) Use the information of parts (a)-(d) to sketch the graph (on axes v(r) vs r). [5|