Exponential Decay Formula
- Introduction
- Formula of Exponential Decay
- Types of Exponential Decay Formula
- Applications of Exponential Decay
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
Exponential decay describes the process of reducing an amount or value by a constant ratio over a period of time. This constant ratio is more commonly expressed as a consistent percentage rate over a period of time.
Exponential decay occurs when something slowly decreases at first, then rapidly diminishes. We use a formula to calculate this decay, which helps us understand how things lose value or decrease in number over time. For instance, imagine you have a pile of `100` apples. Using the exponential decay formula, you can figure out how many apples will be left after a certain amount of time if they decay at a certain rate.
This formula is also handy for determining half-life, which is the time it takes for half of the apples (or whatever you're measuring) to decay or disappear.
Formula Of Exponential Decay
The exponential decay formula helps us understand how things shrink over time. Imagine a snowball melting, or a radioactive substance losing its potency—it's all about that rapid decrease.
The exponential decay formula helps in finding the rapid decrease over a period of time, i.e., the exponential decrease. The exponential decay formula is used to find the population decay, half-life, radioactivity decay, etc.
The general form is \( f(x) = a (1 - r)^x \), where, \( a = \) initial amount, \( r = \) is the rate of decay, \( 1-r = \) decay factor, \( f(x) = \) final amount and \( x = \) time period.
In the above graph `f(x) = (1/4)^x`, the initial amount`(a) = 1` and rate of decay `(r ) = 3/4` meaning decay factor `(1-r) = 1/4`.
Types of Exponential Decay Formula
The process begins with a gradual decrease in quantity, followed by a swift decline. This declining growth is calculated using the exponential decay formula. This formula can appear in various forms, including:
`1`. Basic Exponential Decay Formula:
\(f(x) = ab^x\)
This formula depicts the decay of a quantity \(f(x)\) over time \(x\), where \(a\) is the initial amount, and \(b\) is the decay factor.
`2`. General Exponential Decay Formula:
\(f(x) = a(1 - r)^x\)
Here, \(a\) represents the initial amount, \(r\) signifies the rate of decay, and \(x\) denotes time intervals. When `r` is given as percentage value, it is firs changed to the decimal equivalent and then used in the formula.
`3`. Continuous Exponential Decay Formula:
\(P = P_0e^{-kt}\)
This formula models continuous decay, where \(P\) is the final amount, \(P_0\) is the initial amount, \(k\) is the constant of proportionality, \(e\) is the Euler’s constant and \(t\) denotes time.
Applications of Exponential Decay
`1`. Radioactive Decay: Exponential decay is fundamental in understanding the decay of radioactive isotopes over time, crucial in fields like nuclear physics, radiology, and carbon dating.
`2`. Population Dynamics: It aids in modeling population decline in ecology and demography studies, offering insights into species extinction, population control, and ecosystem stability.
`3`. Medicine: Exponential decay is utilized in pharmacokinetics to determine the rate at which drugs metabolize or decay in the body, guiding dosage regimens and drug effectiveness.
`4`. Finance and Economics: It's employed in financial modelling to analyze depreciation of assets, decrease in stock values, and decay of investment returns over time, influencing investment strategies and economic forecasts.
`5`. Chemistry and Biology: Exponential decay plays a role in understanding chemical reactions, enzyme kinetics, and decay of biological compounds, aiding research in drug development, biochemistry, and molecular biology.
Solved Examples
Example `1`: A sample of a radioactive isotope has an initial mass of `100` grams. The decay factor for this isotope is \( b = 0.8 \). After `5` years, what will be the mass of the sample?
Solution:
The basic exponential decay formula is \( f(x) = ab^x \).
Given: \( a = 100 \) grams, \( b = 0.8 \), \( x = 5 \) years.
Substituting the values into the formula:
\( f(x) = 100 \times (0.8)^5 \)
\( f(x) = 100 \times 0.32768 \)
\( f(x) ≈ 32.768 \)
So, after `5` years, the mass of the sample will be approximately `32.768` grams.
Example `2`: The initial amount of a certain medication in a patient's bloodstream is `200` milligrams. The rate of decay for this medication is \( r = 0.1 \) per hour. How much medication remains after `3` hours?
Solution:
The general exponential decay formula is \( f(x) = a(1 - r)^x \).
Given: \( a = 200 \) milligrams, \( r = 0.1 \), \( x = 3 \) hours.
Substituting the values into the formula:
\( f(x) = 200 \times (1 - 0.1)^3 \)
\( f(x) = 200 \times (0.9)^3 \)
\( f(x) = 200 \times 0.729 \)
\( f(x) ≈ 145.8 \)
So, after `3` hours, approximately `145.8` milligrams of the medication remain in the bloodstream.
Example `3`: A sample of a radioactive isotope has an initial mass of `500` grams. The constant of proportionality for this isotope is \( k = 0.02 \) per year. Find the mass of the sample after `10` years.
Solution:
The continuous exponential decay formula is \( P = P_0 e^{-kt} \).
Given: \( P_0 = 500 \) grams, \( k = 0.02 \), \( t = 10 \) years.
Substituting the values into the formula:
\( P = 500 \times e^{-0.02 \times 10} \)
\( P = 500 \times e^{-0.2} \)
\( P ≈ 500 \times 0.8187 \)
\( P ≈ 409.35 \)
So, after `10` years, the mass of the sample will be approximately `409.35` grams.
Example `4`: A certain chemical pollutant in a lake decays exponentially with a half-life of `2` years. If the initial concentration of the pollutant is `80` parts per million (ppm), what will be the concentration after `6` years?
Solution:
Given: Half-life \( t_{\frac{1}{2}} = 2 \) years, Initial concentration \( a = 80 \) ppm, Time \( x = 6 \) years.
The half-life formula for exponential decay is `f(x) = a \left( \frac{1}{2} \right)^{\frac{x}{t_{1/2}}}`.
Substituting the values:
`f(x) = 80 \left( \frac{1}{2} \right)^{\frac{6}{2}}`
`f(x) = 80 \left( \frac{1}{2} \right)^3`
`f(x) = 80 \times \frac{1}{8}`
`f(x) = 10`
So, after `6` years, the concentration of the pollutant will be `10` ppm.
Example `5`: A car depreciates at a rate of `15%` per year. If its initial value is `$20,000`, what will be its value after `5` years?
Solution:
Given: Initial value \( a = $20,000 \), Rate of depreciation \( r = 0.15 \), Time \( x = 5 \) years.
The formula for financial depreciation is \( f(x) = a(1 - r)^x \).
Substituting the values:
\( f(x) = 20000(1 - 0.15)^5 \)
\( f(x) = 20000(0.85)^5 \)
\( f(x) ≈ 20000 \times 0.443705 \)
\( f(x) ≈ $8,874.10 \)
So, after `5` years, the value of the car will be approximately `$8,874.10`.
Practice Problems
Q`1`. A radioactive substance decays with a decay factor \( b = 0.6 \) per year. If the initial amount of the substance is `500` grams, what will be the amount remaining after `4` years?
- `64.8` grams
- `89` grams
- `98.2` grams
- `103` grams
Answer: a
Q`2`. The population of a city is decreasing with a rate of decay \( r = 0.05 \) per year. If the initial population is `10,000`, what will be the population after `10` years? Round your answer to the nearest whole number.
- `4,000`
- `4,556`
- `5,143`
- `5,987`
Answer: d
Q`3`. A sample of a chemical compound undergoes continuous decay with a constant of proportionality \( k = 0.03 \) per hour. If the initial mass of the compound is `100` grams, what will be the mass after `2` hours?
- `75.5` grams
- `94.18` grams
- `105.12` grams
- `123.5` grams
Answer: b
Q`4`. A certain radioactive isotope has a half-life of `5` years. If the initial amount of the isotope is `200` grams, how much will remain after `15` years?
- `25` grams
- `50` grams
- `75` grams
- `100` grams
Answer: a
Q`5`. A computer depreciates at a rate of `10%` per year. If its initial value is `$1,500`, what will be its value after `3` years?
- `$1,093.5`
- `$945.7`
- `$850.0`
- `$765.2`
Answer: a
Frequently Asked Questions
Q`1`. What is exponential decay?
Answer: Exponential decay refers to the process where a quantity decreases at a rate proportional to its current value. It's characterized by a rapid decrease initially followed by a slower decrease over time.
Q`2`. What is the exponential decay formula used for?
Answer: The exponential decay formula is used to model and predict the decrease of a quantity over time. It finds applications in various fields such as physics, chemistry, biology, finance, and environmental science.
Q`3`. What are the key components of the exponential decay formula?
Answer: The exponential decay formula typically includes parameters such as the initial amount, decay factor or rate, and time. These components help in quantifying the decay process accurately.
Q`4`. How is exponential decay different from exponential growth?
Answer: Exponential decay involves a decrease in quantity over time, whereas exponential growth involves an increase. Both processes follow exponential functions but with different parameters and implications.
Q`5`. Why is exponential decay important in various scientific and practical applications?
Answer: Exponential decay provides a mathematical framework for understanding natural phenomena and practical processes such as decay rates, growth patterns, and time-dependent changes. Its applications range from predicting radioactive decay to managing financial assets and environmental monitoring.