When coughing, the windpipes contract to increase the velocity of air passingthrough the windpipe to help clear mucus. The velocity, u, at which the air flows through thewindpipe depends on the radius, r of the windpipe. If R is the resting radius of the windpipe,then the velocity of air passing through the windpipe satisfies:v(r) = Ar²(R – r),where A is a constant dependent on the strength of the diaphram muscles. Find thederivative of the velocity function: (Note that your answers in the questions below mayinclude A and R.)v'(r) =Find the value of r that maximizes the velocity of air.Imax =Determine the maximum velocity of the air flowing through the windpipe.v(rmax) =

We need to find out the v'(r), value of r that maximizes the velocity of air and v(rmax).

Answered: When coughing, the windpipes contract…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The density of air changes with height. Under some conditions density p, depends on height z, and temperature T according to the following equation where Po and A are both constants. A meteorological balloon ascends (i.e., starts at z = 1 and gains height) over the course of several hours. Complete parts (a) and (b) below. Az P(z,T) = Po e ..... dz v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, (a) Assuming that the balloon ascends at a speed v (i.e., dt as measured by the weather balloon. Choose the correct answer below. dp Az dT O A. dt T2 dt dp %3D dt Az dT dp С. dt %3D + T dt dp Az) dT O D. dt T2) dt (b) Assume that v = 1, Po = 1, and A = 1 and that when t= 0, T= 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = 0, and solve this…
EXERCISE 2.2 An automobile follows a circular road whose radius is 50 m. Let x and y respectively denote the eastern and northern directions, with origin at the center of the circle. Suppose the vehicle starts from rest at x = 50 m heading north, and its speed depends on the distance s it travels according to v=0.5s-0.0025s², where s is measured in meters and v is in meters per second. It is known that the tires will begin to skid when the total acceleration of the vehicle is 0.6g. Where will the automobile be and how fast will it be going when it begins to skid? Describe the position in terms of the angle of the radial line relative to the x axis.
The volume of water in a cylindrical tank is given by: V = лr²h where V is the volume of water in m³, r is the radius of the tank in metres, and h is the height of the water in metres. For a certain tank, the radius is 3.0 m. This is a property of the tank, so it is a constant. Water is filling the tank, so the height and volume are constantly changing hence h and V are variables. At a certain point in time, the height of the tank is increasing by 0.21 metres/hour and the height is 29 metres. Determine the rate of change of volume (in units of m³/h) at this point in time. Give your answer to three decimal places. Hint 1: You are being asked to find dv/dt. Hint 2: dh/dt = 0.21 m/h
A basket of flowers of mass 3 kg is placed on a flat grassy slope that makes an angle θ with the horizontal. The coefficient of static friction between the basket and the slope is 0.45 and the basket is on the point of slipping down the slope. Model the basket of flowers as a particle and the grassy slope as a plane. Take the magnitude of the acceleration due to gravity, g, to be 9.8 m s−2 Express the forces in component form, in terms of θ and unknown magnitudes where appropriate. Write down the equilibrium condition for the basket and hence show that tan θ = 0.45. Determine the angle, in degrees, that the slope makes with the horizontal.
A sheet of water of uniform thickness (h = 0.03 m) flows from the device shown in the figure below. The water enters vertically through the inlet pipe and exits horizontally with a speed that varies linearly from 0 to 11 m/s along the 0.2-m length of the slit. Determine the y component of anchoring force necessary to hold this device stationary. FAY = 0 m/s- i 0.2m 0.03m N 11m/s

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