A First Course in Differential Equations with Modeling Applications (MindTap Course List)
11th Edition
ISBN: 9781305965720
Author: Dennis G. Zill
Publisher: Cengage Learning
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Chapter 3, Problem 9RE
Suppose that as a body cools, the temperature of the surrounding medium increases because it completely absorbs the heat being lost by the body. Let T(t) and Tm(t) be the temperatures of the body and the medium at time t, respectively.
If the initial temperature of the body is T1 and the initial temperature of the medium is T2, then it can be shown in this case that Newton’s law of cooling is dT/dt = k(T − Tm), k < 0, where Tm = T2 + B(T1 − T), B > 0 is a constant.
- (a) The foregoing DE is autonomous. Use the phase portrait concept of Section 2.1 to determine the limiting value of the temperature T(t) as t → ∞. What is the limiting value of Tm(t) as t → ∞?
- (b) Verify your answers in part (a) by actually solving the differential equation.
- (c) Discuss a physical interpretation of your answers in part (a).
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Chapter 3 Solutions
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
Ch. 3.1 - The population of a community is known to increase...Ch. 3.1 - Suppose it is known that the population of the...Ch. 3.1 - The population of a town grows at a rate...Ch. 3.1 - The population of bacteria in a culture grows at a...Ch. 3.1 - The radioactive isotope of lead, Pb-209, decays at...Ch. 3.1 - Initially 100 milligrams of a radioactive...Ch. 3.1 - Determine the half-life of the radioactive...Ch. 3.1 - Consider the initial-value problem dA/dt = kA,...Ch. 3.1 - When a vertical beam of light passes through a...Ch. 3.1 - When interest is compounded continuously, the...
Ch. 3.1 - Carbon Dating Archaeologists used pieces of burned...Ch. 3.1 - The Shroud of Turin, which shows the negative...Ch. 3.1 - Newtons Law of Cooling/Warming A thermometer is...Ch. 3.1 - A thermometer is taken from an inside room to the...Ch. 3.1 - A small metal bar, whose initial temperature was...Ch. 3.1 - Two large containers A and B of the same size are...Ch. 3.1 - A thermometer reading 70 F is placed in an oven...Ch. 3.1 - At t = 0 a sealed test tube containing a chemical...Ch. 3.1 - A dead body was found within a closed room of a...Ch. 3.1 - The rate at which a body cools also depends on its...Ch. 3.1 - A tank contains 200 liters of fluid in which 30...Ch. 3.1 - Solve Problem 21 assuming that pure water is...Ch. 3.1 - A large tank is filled to capacity with 500...Ch. 3.1 - In Problem 23, what is the concentration c(t) of...Ch. 3.1 - Solve Problem 23 under the assumption that the...Ch. 3.1 - Determine the amount of salt in the tank at time t...Ch. 3.1 - A large tank is partially filled with 100 gallons...Ch. 3.1 - In Example 5 the size of the tank containing the...Ch. 3.1 - A 30-volt electromotive force is applied to an...Ch. 3.1 - Prob. 30ECh. 3.1 - A 100-volt electromotive force is applied to an...Ch. 3.1 - A 200-volt electromotive force is applied to an...Ch. 3.1 - An electromotive force E(t)={120,0t200,t20 is...Ch. 3.1 - An LR-series circuit has a variable inductor with...Ch. 3.1 - Air Resistance In (14) of Section 1.3 we saw that...Ch. 3.1 - How High?No Air Resistance Suppose a small...Ch. 3.1 - How High?Linear Air Resistance Repeat Problem 36,...Ch. 3.1 - Skydiving A skydiver weighs 125 pounds, and her...Ch. 3.1 - Rocket Motion Suppose a small single-stage rocket...Ch. 3.1 - Rocket MotionContinued In Problem 39 suppose of...Ch. 3.1 - Evaporating Raindrop As a raindrop falls, it...Ch. 3.1 - Prob. 42ECh. 3.1 - Prob. 43ECh. 3.1 - Constant-Harvest model A model that describes the...Ch. 3.1 - Drug Dissemination A mathematical model for the...Ch. 3.1 - Memorization When forgetfulness is taken into...Ch. 3.1 - Heart Pacemaker A heart pacemaker, shown in Figure...Ch. 3.1 - Sliding Box (a) A box of mass m slides down an...Ch. 3.1 - Sliding Box—Continued
In Problem 48 let s(t) be...Ch. 3.1 - What Goes Up (a) It is well known that the model...Ch. 3.2 - The number N(t) of supermarkets throughout the...Ch. 3.2 - The number N(t) of people in a community who are...Ch. 3.2 - A model for the population P(t) in a suburb of a...Ch. 3.2 - Census data for the United States between 1790 and...Ch. 3.2 - (a) If a constant number h of fish are harvested...Ch. 3.2 - Investigate the harvesting model in Problem 5 both...Ch. 3.2 - Repeat Problem 6 in the case a = 5, b = 1, h = 7.Ch. 3.2 - (a) Suppose a = b = 1 in the Gompertz differential...Ch. 3.2 - Two chemicals A and B are combined to form a...Ch. 3.2 - Solve Problem 9 if 100 grams of chemical A is...Ch. 3.2 - Leaking cylindrical tank A tank in the form of a...Ch. 3.2 - Leaking cylindrical tank—continued When friction...Ch. 3.2 - Leaking Conical Tank A tank in the form of a...Ch. 3.2 - Inverted Conical Tank Suppose that the conical...Ch. 3.2 - Air Resistance A differential equation for the...Ch. 3.2 - How High?Nonlinear Air Resistance Consider the...Ch. 3.2 - That Sinking Feeling (a) Determine a differential...Ch. 3.2 - Solar Collector The differential equation...Ch. 3.2 - Tsunami (a) A simple model for the shape of a...Ch. 3.2 - Evaporation An outdoor decorative pond in the...Ch. 3.2 - Doomsday equation Consider the differential...Ch. 3.2 - Doomsday or extinction Suppose the population...Ch. 3.2 - Skydiving A skydiver is equipped with a stopwatch...Ch. 3.2 - Prob. 27ECh. 3.2 - Old Man River In Figure 3.2.8(a) suppose that the...Ch. 3.2 - Prob. 29ECh. 3.2 - Prob. 30ECh. 3.2 - Prob. 31ECh. 3.2 - Prob. 32ECh. 3.2 - Time Drips By The clepsydra, or water clock, was a...Ch. 3.2 - (a) Suppose that a glass tank has the shape of a...Ch. 3.2 - Prob. 35ECh. 3.3 - We have not discussed methods by which systems of...Ch. 3.3 - In Problem 1 suppose that time is measured in...Ch. 3.3 - Use the graphs in Problem 2 to approximate the...Ch. 3.3 - Construct a mathematical model for a radioactive...Ch. 3.3 - Potassium-40 Decay The chemical element potassium...Ch. 3.3 - Potassium-Argon Dating The knowledge of how K-40...Ch. 3.3 - Consider two tanks A and B, with liquid being...Ch. 3.3 - Use the information given in Figure 3.3.6 to...Ch. 3.3 - Two very large tanks A and B are each partially...Ch. 3.3 - Three large tanks contain brine, as shown in...Ch. 3.3 - Consider the Lotka-Volterra predator-prey model...Ch. 3.3 - Show that a system of differential equations that...Ch. 3.3 - Determine a system of first-order differential...Ch. 3.3 - Prob. 16ECh. 3.3 - SIR Model A communicable disease is spread...Ch. 3.3 - Prob. 18ECh. 3.3 - Prob. 19ECh. 3.3 - Prob. 20ECh. 3.3 - Mixtures Solely on the basis of the physical...Ch. 3.3 - Newtons Law of Cooling/Warming As shown in Figure...Ch. 3 - Answer Problems 1 and 2 without referring back to...Ch. 3 - Answer Problems 1 and 2 without referring back to...Ch. 3 - Prob. 3RECh. 3 - Air containing 0.06% carbon dioxide is pumped into...Ch. 3 - tzi the Iceman In September of 1991 two German...Ch. 3 - Prob. 6RECh. 3 - Prob. 7RECh. 3 - Suppose a cell is suspended in a solution...Ch. 3 - Suppose that as a body cools, the temperature of...Ch. 3 - According to Stefans law of radiation the absolute...Ch. 3 - Suppose an RC-series circuit has a variable...Ch. 3 - A classical problem in the calculus of variations...Ch. 3 - A model for the populations of two interacting...Ch. 3 - Initially, two large tanks A and B each hold 100...Ch. 3 - Prob. 15RECh. 3 - When all the curves in a family G(x, y, c1) = 0...Ch. 3 - Prob. 17RECh. 3 - Prob. 18RECh. 3 - Prob. 19RECh. 3 - Sawing Wood A long uniform piece of wood (cross...Ch. 3 - Solve the initial-value problem in Problem 20 when...Ch. 3 - Prob. 22RE
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