Question

Consider the models of exponential decay $f(t)=(2+c) e^{-k_{1} t}$ and $g(t)=c e^{-k_{2} t} .$ Suppose that $f(1)=g(1),$ what is the relationship between $k_{1}$ and $k_{2} ?$

Name: Problem 57, Chapter 3: Precalculus
Uploaded: 2020-07-06T01:59:56-08:00
Duration: 3 min 22 s
Channel: Tp Sarathy
Description: Consider the models of exponential decay f(t)=(2+c) e^-k1 t and g(t)=c e^-k2 t . Suppose that f(1)=g(1), what is the relationship between k1 and k2 ?

   Consider the models of exponential decay $f(t)=(2+c) e^{-k_{1} t}$ and $g(t)=c e^{-k_{2} t} .$ Suppose that $f(1)=g(1),$ what is the relationship between $k_{1}$ and $k_{2} ?$

Precalculus

Cynthia Young 2nd Edition

Chapter 3, Problem 57

Instant Answer

Solved by Expert Tp Sarathy

07/06/2020

Step 1

We substitute $t=1$ into the functions $f(t)$ and $g(t)$ to get the equations $f(1)=(2+c) e^{-k_{1}}$ and $g(1)=c e^{-k_{2}}$. Show more…

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Consider the models of exponential decay $f(t)=(2+c) e^{-k_{1} t}$ and $g(t)=c e^{-k_{2} t} .$ Suppose that $f(1)=g(1),$ what is the relationship between $k_{1}$ and $k_{2} ?$

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Key Concepts

Logarithmic Transformation in Solving Exponential Equations

Solving equations involving exponentials often requires taking logarithms to linearize the exponential expressions. This transformation leverages the properties of logarithms to isolate and determine relationships between the decay rate parameters.

Exponential Decay Functions

Exponential decay functions model processes where a quantity decreases at a rate proportional to its current value, typically represented by an equation of the form y(t) = A·e^(–kt). The constant A signifies the initial value and k, known as the decay rate, determines how quickly the quantity decreases over time.

Parameter Equivalence in Models

When two exponential decay models are set to be equal at a specific time, equating their expressions creates a constraint that links their parameters. This process ensures consistency between the models at that point, leading to relationships among the constants and decay rates.

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