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.

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.