Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.4.30

Chapter 9, Problem 9.4.30

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.

A pot of boiling soup (100°C) is put in a cellar with a temperature of 10°C. After 30 minutes, the soup has cooled to 80°C. When will the temperature of the soup reach 30°C

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Identify the variables and constants in Newton's Law of Cooling. Let \(T(t)\) be the temperature of the soup at time \(t\) (in minutes), \(T_a\) be the ambient temperature (10°C), and \(T_0\) be the initial temperature of the soup (100°C).

Write the differential equation representing Newton's Law of Cooling: \(\frac{dT}{dt} = -k (T - T_a)\), where \(k\) is a positive constant representing the cooling rate.

Solve the differential equation by separating variables or using an integrating factor. The general solution has the form: \(T(t) = T_a + (T_0 - T_a) e^{-k t}\).

Use the given information that after 30 minutes, the temperature is 80°C to find the constant \(k\). Substitute \(t=30\), \(T(30) = 80\), \(T_a = 10\), and \(T_0 = 100\) into the solution and solve for \(k\).

Once \(k\) is found, set \(T(t) = 30\) and solve for \(t\) to find when the soup reaches 30°C.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton’s Law of Cooling

Newton’s Law of Cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature. Mathematically, it is expressed as a first-order differential equation, which models how the temperature approaches the surrounding temperature over time.

Solving First-Order Differential Equations

To find the temperature function, you solve the first-order linear differential equation derived from Newton’s Law of Cooling. This involves separating variables or using an integrating factor to obtain a general solution that describes temperature as a function of time.

Applying Initial Conditions and Solving for Time

After finding the general temperature function, initial conditions like the starting temperature and temperature at a specific time are used to determine constants. Then, to find when the temperature reaches a certain value, you solve the equation for time by substituting the target temperature.

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p'(x) = 16/x⁹ - 5 + 14x⁶

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r=0.2, K=300, P₀=50

Textbook Question

Stability of Euler's method Consider the initial value problem y′(t) = −ay, y(0) = 1 where a > 0; it has the exact solution y(t) = e⁻ᵃᵗ, which is a decreasing function.

a. Show that Euler's method applied to this problem with time step h can be written u₀ = 1, uₖ₊₁ = (1 − ah)uₖ for k = 0, 1, 2, ...

b. Show by substitution that uₖ = (1 − ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

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Textbook Question

Explain how to solve a separable differential equation of the form

g(t)y'(y) = h(t)

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Textbook Question

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.

An iron rod is removed from a blacksmith’s forge at a temperature of 900°C . Assume k=0.02 and the rod cools in a room with a temperature of 30°C When does the temperature of the rod reach 100°C?

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Textbook Question

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

e⁴ᵗy'(t) = 5

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