Textbook Question
Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.
a. Sketch a solution on the direction field with the initial condition y(0)=1.
Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 9.R.30c
Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.
c. When does the temperature of the coffee reach 50°C?
Verified step by step guidance1
Identify the variables and constants in Newton's Law of Cooling. Let \(T(t)\) be the temperature of the coffee at time \(t\) (in minutes), \(T_s = 25^\circ C\) be the surrounding room temperature, and \(T_0 = 80^\circ C\) be the initial temperature of the coffee at \(t=0\).
Write the general form of Newton's Law of Cooling:
\[T(t) = T_s + (T_0 - T_s) e^{-kt}\]
where \(k\) is a positive constant that depends on the cooling rate.
Use the given information that after 5 minutes, the temperature is 60°C to find the constant \(k\). Substitute \(t=5\), \(T(5) = 60\), \(T_s = 25\), and \(T_0 = 80\) into the equation:
\[60 = 25 + (80 - 25) e^{-5k}\]
Solve this equation for \(k\) (do not calculate the numerical value, just isolate \(k\)).
Set \(T(t) = 50\) to find the time \(t\) when the coffee reaches 50°C. Substitute \(T(t) = 50\), \(T_s = 25\), \(T_0 = 80\), and the expression for \(k\) into the formula:
\[50 = 25 + (80 - 25) e^{-kt}\]
Solve the equation from step 4 for \(t\) by isolating the exponential term and then taking the natural logarithm. This will give you the time when the coffee reaches 50°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 describes how the temperature of an object changes over time as it approaches the ambient temperature. It states that the rate of change of the temperature is proportional to the difference between the object's temperature and the surrounding temperature.
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Newton's Law of Cooling
Exponential Decay Model
The temperature change in Newton’s Law of Cooling follows an exponential decay model, where the temperature difference decreases exponentially over time. This model is expressed as T(t) = T_env + (T_initial - T_env) * e^(-kt), where k is a positive constant.
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Solving for Time in Cooling Problems
To find when the temperature reaches a certain value, you substitute the known temperatures into the cooling formula and solve for time t. This involves isolating t in the exponential equation, often using logarithms to handle the exponential term.
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Related Practice
Textbook Question
Euler’s metho d Consider the initial value problem y′(t)=1/2y,y(0)=1.
a. Use Euler’s method with Δt=0.1 to compute approximations to y(0.1) and y(0.2).
Textbook Question
Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added at t=0. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells.
c. After how many hours does the population reach half of the carrying capacity
Textbook Question
Logistic growth The population of a rabbit community is governed by the initial value problem
P′(t) = 0.2 P (1 − P/1200), P(0) = 50
a. Find the equilibrium solutions.
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Textbook Question
Euler’s method Consider the initial value problem y′(t)=1/2y,y(0)=1.
b. Use Euler’s method with Δt=0.05 to compute approximations to y(0.1) and y(0.2).
Textbook Question
Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.
d. Complete the following sentence. The solution of the differential equation with the initial condition y(0)=A, where A is a real number, approaches the line _____ as t→∞.