Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.2.12

Chapter 7, Problem 7.2.12

After the introduction of foxes on an island, the number of rabbits on the island decreases by 4.5% per month. If y(t) equals the number of rabbits on the island t months after foxes were introduced, find the rate constant k for the exponential decay function y(t) = y₀eᵏᵗ.

Verified step by step guidance

Recognize that the problem describes an exponential decay process where the quantity decreases by a fixed percentage each month. The general form of the function is given as $y(t) = y_0 e^{k t}$, where $k$ is the rate constant we need to find.

Understand that a 4.5% decrease per month means that after 1 month, the population is 95.5% of the original, or mathematically, $y(1) = 0.955 y_0$.

Substitute $t = 1$ into the exponential decay formula: $y(1) = y_0 e^{k \cdot 1} = y_0 e^{k}$.

Set the two expressions for $y(1)$ equal to each other: $y_0 e^{k} = 0.955 y_0$. Since $y_0$ is not zero, divide both sides by $y_0$ to get $e^{k} = 0.955$.

Take the natural logarithm of both sides to solve for $k$: $k = \ln(0.955)$. This gives the rate constant for the exponential decay.

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Key Concepts

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

Exponential Decay Function

An exponential decay function models quantities that decrease at a rate proportional to their current value. It is generally expressed as y(t) = y₀e^{kt}, where k is a negative constant representing the decay rate, y₀ is the initial amount, and t is time.

Rate Constant (k) in Exponential Decay

The rate constant k determines how quickly the quantity decreases over time. For decay, k is negative, and its magnitude relates to the percentage decrease per time unit. Finding k involves converting the given percentage decrease into a continuous decay rate.

Converting Percentage Decrease to Continuous Decay Rate

A percentage decrease per time period can be converted to the continuous decay rate k by using the formula k = ln(1 - r), where r is the decimal form of the percentage decrease. This links discrete percentage changes to the continuous exponential model.

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