Textbook Question
Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.4.37
37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?
Verified step by step guidance1
Identify the decay model: The amount of plutonium remaining after time \(t\) can be modeled by the exponential decay formula \(A(t) = A_0 \cdot e^{-kt}\), where \(A_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is time in years.
Use the half-life to find the decay constant \(k\): Since the half-life \(T_{1/2}\) is 24,360 years, use the relation \(k = \frac{\ln(2)}{T_{1/2}} = \frac{\ln(2)}{24360}\) to calculate \(k\).
Set up the equation for 80% decay: If 80% of the plutonium decays, 20% remains. So, \(A(t) = 0.2 \times A_0\). Substitute into the decay formula to get \(0.2 A_0 = A_0 \cdot e^{-kt}\).
Simplify the equation: Cancel \(A_0\) from both sides to get \(0.2 = e^{-kt}\). Then take the natural logarithm of both sides to solve for \(t\): \(\ln(0.2) = -kt\).
Solve for \(t\): Rearrange to find \(t = -\frac{\ln(0.2)}{k}\). Substitute the value of \(k\) from step 2 to express \(t\) in terms of known quantities.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Decay
Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In radioactive decay, the amount of substance decreases exponentially over time, modeled by the formula N(t) = N_0 * e^(-kt), where k is the decay constant.
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Half-Life
Half-life is the time required for half of a radioactive substance to decay. It is a constant unique to each isotope and relates to the decay constant by the formula t_(1/2) = ln(2)/k. Knowing the half-life allows calculation of the decay constant and prediction of remaining substance over time.
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Decay Constant and Time Calculation
The decay constant (k) quantifies the rate of decay and is used to find the time for a certain fraction of the substance to decay. By rearranging the decay formula, time can be calculated as t = -(1/k) * ln(N(t)/N_0), enabling determination of how long it takes for a specific percentage of the isotope to decay.
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Related Practice
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Verify the integration formulas in Exercises 111–114.
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57. ∫(from 1 to 2)cosh(ln t)/t dt
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In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
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In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(cost+lnt)