4. a 200-gallon initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 0.25 pounds of salt per gallon is added to the tank at a rate of 4 gallons per minute, and the resulting mixture is drained out at 2 gallons per minute. Find the quantity of salt in the tank as it is about to overflow. 5. A boat weights 64,000 pounds. Its propeller produces a constant thrust of 50,000 pounds and the water exerts a resistive force with magnitude proportional to the speed, with k = 2000 pounds per foot. Assuming that the boat starts from rest, find its velocity as a function of time as well as the terminal velocity. 6. Suppose that T(t) is the temperature T of an object at time t, Tm is the temperature of the surrounding medium which is constant, and the rate of the change of the temperature is proportional to the differnce of the temperature of the object and of the surrounding medium. Setup a differential equation for this situation and finally solve for the general solution. 7. Suppose that yi is a solution the associated homogeneous differntial equation of y′+ p(x)y = f(x). Show that u' f(x) Y1 where the u comes from assuming the general solution of the non-homogenous differential equation is y = uy
4. a 200-gallon initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 0.25 pounds of salt per gallon is added to the tank at a rate of 4 gallons per minute, and the resulting mixture is drained out at 2 gallons per minute. Find the quantity of salt in the tank as it is about to overflow. 5. A boat weights 64,000 pounds. Its propeller produces a constant thrust of 50,000 pounds and the water exerts a resistive force with magnitude proportional to the speed, with k = 2000 pounds per foot. Assuming that the boat starts from rest, find its velocity as a function of time as well as the terminal velocity. 6. Suppose that T(t) is the temperature T of an object at time t, Tm is the temperature of the surrounding medium which is constant, and the rate of the change of the temperature is proportional to the differnce of the temperature of the object and of the surrounding medium. Setup a differential equation for this situation and finally solve for the general solution. 7. Suppose that yi is a solution the associated homogeneous differntial equation of y′+ p(x)y = f(x). Show that u' f(x) Y1 where the u comes from assuming the general solution of the non-homogenous differential equation is y = uy
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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Transcribed Image Text:4. a 200-gallon initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 0.25
pounds of salt per gallon is added to the tank at a rate of 4 gallons per minute, and the resulting mixture
is drained out at 2 gallons per minute. Find the quantity of salt in the tank as it is about to overflow.
5. A boat weights 64,000 pounds. Its propeller produces a constant thrust of 50,000 pounds and the
water exerts a resistive force with magnitude proportional to the speed, with k = 2000 pounds per foot.
Assuming that the boat starts from rest, find its velocity as a function of time as well as the terminal
velocity.
6. Suppose that T(t) is the temperature T of an object at time t, Tm is the temperature of the surrounding
medium which is constant, and the rate of the change of the temperature is proportional to the differnce
of the temperature of the object and of the surrounding medium. Setup a differential equation for this
situation and finally solve for the general solution.
7. Suppose that yi is a solution the associated homogeneous differntial equation of y′+ p(x)y = f(x). Show
that
u'
f(x)
Y1
where the u comes from assuming the general solution of the non-homogenous differential equation is
y = uy
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