### Problem StatementIf a substance decomposes at a rate proportional to the amount of the substance present, and the amount decreases from 60 ounces to 15 ounces in 2 hours, find the constant of proportionality.### Options- $ \large -\frac{1}{2} $- $ \large -\ln(2) $- $ \large -\frac{1}{4} $- $ \large \ln \left( \frac{1}{4} \right) $#### ExplanationThis question is designed to test the understanding of decay processes which follow first-order kinetics, and how to derive the constant of proportionality using given data points. In this case, the decrease in the amount of substance is used to determine the proportionality constant (decay constant) using logarithmic relationships.The proportionality follows an exponential decay model, typically represented by the equation:\[ A(t) = A_0 e^{-kt} \]where:- $ A(t) $ is the amount of substance at time $ t $,- $ A_0 $ is the initial amount of substance,- $ k $ is the decay constant,- $ t $ is the time. Given the initial and remaining amounts, along with the time elapsed, students are expected to calculate $ k $.

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If a substance decomposes at a rate proportional to the amount of the substance present, and the amount decreases from 60 ounces to 15 ounces in 2 hours, find the constant of proportionality. O – In(2) O In()

If a substance decomposes at a rate proportional to the amount of the substance present, and the amount decreases from 60 ounces to 15 ounces in 2 hours, find the constant of proportionality. O – In(2) O In()

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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### Problem Statement
If a substance decomposes at a rate proportional to the amount of the substance present, and the amount decreases from 60 ounces to 15 ounces in 2 hours, find the constant of proportionality.

### Options
- $ \large -\frac{1}{2} $
- $ \large -\ln(2) $
- $ \large -\frac{1}{4} $
- $ \large \ln \left( \frac{1}{4} \right) $

#### Explanation
This question is designed to test the understanding of decay processes which follow first-order kinetics, and how to derive the constant of proportionality using given data points. In this case, the decrease in the amount of substance is used to determine the proportionality constant (decay constant) using logarithmic relationships.

The proportionality follows an exponential decay model, typically represented by the equation:
\[ A(t) = A_0 e^{-kt} \]
where:
- $ A(t) $ is the amount of substance at time $ t $,
- $ A_0 $ is the initial amount of substance,
- $ k $ is the decay constant,
- $ t $ is the time.

Given the initial and remaining amounts, along with the time elapsed, students are expected to calculate $ k $.

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