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What are Applications of Differential Equations in Decay Models?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Applications of differential equations in decay models help us understand how things decrease over time. They describe situations where the rate at which something reduces is proportional to the amount present at that moment. This is useful for predicting how quickly substances break down or quantities diminish.
Simple Example
Quick Example
Imagine you have 1000 rupees in a special bank account that loses 10% of its remaining money every year due to some fee. A differential equation helps us calculate exactly how much money will be left after 1 year, 2 years, or even 5 years, showing how it decays over time.
Worked Example
Step-by-Step
Problem: A radioactive substance decays such that its rate of decay is proportional to the amount present. If 100 grams are present initially and after 1 hour, 90 grams remain, find the amount remaining after 3 hours.
Step 1: Set up the differential equation for decay. Let N be the amount of substance and t be time. The rate of decay is dN/dt = -kN, where k is the decay constant.
---Step 2: Solve the differential equation. Integrating both sides gives ln(N) = -kt + C, which means N = e^(-kt+C) = e^C * e^(-kt). Let A = e^C, so N = A * e^(-kt).
---Step 3: Use initial conditions to find A. At t=0, N=100. So, 100 = A * e^(0), which means A = 100. The equation becomes N = 100 * e^(-kt).
---Step 4: Use the second condition to find k. At t=1 hour, N=90. So, 90 = 100 * e^(-k*1). This gives 0.9 = e^(-k). Taking natural log of both sides, ln(0.9) = -k. So, k = -ln(0.9) which is approximately 0.10536.
---Step 5: Now the full equation is N = 100 * e^(-0.10536t).
---Step 6: Find the amount remaining after 3 hours. Substitute t=3 into the equation: N = 100 * e^(-0.10536 * 3).
---Step 7: Calculate N = 100 * e^(-0.31608) = 100 * 0.7288. So, N is approximately 72.88 grams.
Answer: After 3 hours, approximately 72.88 grams of the substance will remain.
Why It Matters
Understanding decay models is super important in many fields. Doctors use them to figure out how fast medicines leave our body, engineers use them to predict how long materials will last, and scientists use them to study climate change or the age of ancient artifacts. This knowledge can lead to exciting careers in medicine, environmental science, or research!
Common Mistakes
MISTAKE: Forgetting the negative sign in the decay equation (dN/dt = kN) | CORRECTION: Always use a negative sign (dN/dt = -kN) because decay means the quantity is decreasing, so the rate of change is negative.
MISTAKE: Confusing half-life with the decay constant 'k' | CORRECTION: Half-life is the time it takes for half the substance to decay, while 'k' is the proportionality constant in the differential equation. They are related but not the same value.
MISTAKE: Not correctly applying initial conditions to find the constant 'A' (initial amount) | CORRECTION: Always substitute t=0 and the initial amount N_0 into the general solution N = A * e^(-kt) to find the exact value of A.
Practice Questions
Try It Yourself
QUESTION: A radioactive isotope decays at a rate proportional to its amount. If 200 grams are present initially, and after 2 hours, 180 grams remain, how much will be left after 4 hours? | ANSWER: Approximately 162 grams
QUESTION: The population of a certain species of fish in a polluted lake decreases at a rate proportional to its current population. If the population halves every 5 years, and there are initially 10,000 fish, how many fish will be left after 15 years? | ANSWER: 1250 fish
QUESTION: A new medicine breaks down in the body such that 15% of the initial dose is eliminated every hour. If a patient takes a 500mg dose, what is the half-life of the medicine in their body (time for half the dose to remain)? | ANSWER: Approximately 4.25 hours
MCQ
Quick Quiz
Which of the following describes a decay model?
A bank account growing with compound interest.
The number of bacteria doubling every hour.
The amount of a radioactive substance decreasing over time.
The speed of a car increasing steadily.
The Correct Answer Is:
C
Decay models describe situations where a quantity decreases over time. Radioactive decay is a classic example of this, where the substance reduces continuously.
Real World Connection
In the Real World
In India, ISRO (Indian Space Research Organisation) uses principles of decay models when designing power sources for satellites. They need to understand how long a radioactive isotope will provide power before decaying too much. Also, in food science, these models help predict the shelf-life of packaged foods, ensuring they are safe to eat for a certain period.
Key Vocabulary
Key Terms
DECAY: The process of decreasing or breaking down over time | DIFFERENTIAL EQUATION: An equation involving a function and its derivatives, showing how a quantity changes | HALF-LIFE: The time it takes for half of a substance to decay | PROPORTIONAL: When two quantities change at the same rate, maintaining a constant ratio | DECAY CONSTANT: The constant 'k' that determines the rate of decay in the differential equation.
What's Next
What to Learn Next
Next, you can explore 'Applications of Differential Equations in Growth Models'. This builds directly on decay models but describes situations where quantities increase over time, like population growth or compound interest. It's super exciting to see how similar math can explain opposite real-world changes!
