Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The differential equation y′+2y=t is first-order, linear, and separable.
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 9.4.38a
Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.
a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).
Verified step by step guidance1
Start with Newton's Law of Cooling, which states that the temperature \( T(t) \) of the object at time \( t \) satisfies the differential equation:
\[\frac{dT}{dt} = -k (T - A)\]
where \( k > 0 \) is a constant and \( A \) is the ambient temperature.
Solve the differential equation by separating variables or recognizing it as a first-order linear ODE. The general solution is:
\[T(t) = A + (T_0 - A) e^{-k t}\]
where \( T_0 \) is the initial temperature at \( t = 0 \).
Define \( t_{1/2} \) as the time when the temperature reaches half of the initial temperature, i.e., when \( T(t_{1/2}) = \frac{T_0}{2} \). Substitute this into the solution:
\[\frac{T_0}{2} = A + (T_0 - A) e^{-k t_{1/2}}\]
Isolate the exponential term:
\[\frac{T_0}{2} - A = (T_0 - A) e^{-k t_{1/2}}\]
Then divide both sides by \( T_0 - A \):
\[\frac{\frac{T_0}{2} - A}{T_0 - A} = e^{-k t_{1/2}}\]
Take the natural logarithm of both sides to solve for \( t_{1/2} \):
\[-k t_{1/2} = \ln \left( \frac{\frac{T_0}{2} - A}{T_0 - A} \right)\]
Rearranging gives:
\[t_{1/2} = -\frac{1}{k} \ln \left( \frac{T_0 - 2A}{2 (T_0 - A)} \right)\]
which is the desired expression.
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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, this is expressed as dT/dt = -k(T - A), where k > 0 is a constant. This law models how objects cool or warm over time toward the surrounding temperature.
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Newton's Law of Cooling
Solving First-Order Linear Differential Equations
The temperature function T(t) satisfies a first-order linear differential equation. Solving it involves separating variables or using an integrating factor to find T as a function of time. This solution typically involves an exponential decay term reflecting how temperature approaches ambient temperature over time.
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Logarithmic Manipulation in Solving for Time
To find the cooling time t₁/₂ when the temperature reaches T₀/2, one must solve the exponential equation for t. This requires isolating t by taking the natural logarithm of both sides, using properties of logarithms to simplify the expression, and expressing t₁/₂ in terms of known quantities like T₀, A, and k.
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5:02Solving Logarithmic Equations
Related Practice
Textbook Question
{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.
a. Find the solution of the initial value problem.
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Textbook Question
42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
a. Find the general solution of the equation.
e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.
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Textbook Question
52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.
where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.
a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.
Textbook Question
Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.
c. At what time does the mass of sugar reach 95% of its steady-state level?
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