Calculus Answers

a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough.

Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity y can be modeled by the equation , , where is the rest radius of the trachea in centimeters and is a positive constant whose value depends in part on the length of the trachea. Show that is greatest when that is, when the trachea is about contracted. The remarkable fact is that ray photographs confirm that the trachea contracts about this much during a cough.

b. Take to be and to be and graph over the interval . Compare what you see with the claim that is at a maximum.

this question needs to be done pn matlab. Can anybody do this entire question on matlab and send the graph and graph codes

Currently the sowing of wheat is taking place in Pakistan till December, the harvesting season will

begin in March. So, the farmers wants to build a silo in the form of cylinder to keep the wheat inside

the silo after harvesting. For this purpose, they have to built silo of different sizes having 2000 cubic

units and 4000 cubic units. Moreover, the top of the cylinder is hemi-sphere. The cost of construction

of per unit surface area is thrice as great for the hemisphere as it is for the cylindrical sidewall.

Determine the dimensions to be used and cost of construction is to be kept to a minimum. Neglect the

thickness of the silo and waste in construction. Finally, use MATLAB to write a program which will

provide you the optimal dimensions subject to the constraint of cost. The program will take dimensions

of the Silo as input and return the cost and quantity of each size.

Trace the curve x=a(theta + sin theta ) , y= a(1+ cos theta ). State the properties you use tracing it , also?