Explanation:
Exponential decay is a mathematical concept that describes the decrease in a quantity over time. It is characterized by a constant ratio of decrease, where the amount decreases by the same percentage in each unit of time. This concept is applicable in various fields, including mathematics, physics, biology, and finance.
One real-life example of exponential decay is radioactive decay. Radioactive substances, such as uranium or carbon-14, undergo a process where their atomic nuclei spontaneously break down, emitting radiation in the form of particles or energy. The rate at which this decay occurs follows an exponential decay model.
For instance, let's consider a sample of radioactive material with a half-life of 10 years. The half-life is the time it takes for half of the radioactive atoms in the sample to decay. After the first 10 years, half of the atoms will have decayed, leaving only half of the original amount. After another 10 years, half of the remaining atoms will decay, leaving only a quarter of the original amount. This process continues, with each successive half-life reducing the amount of radioactive material by half.
Another example of exponential decay can be found in population growth. When a population exceeds its carrying capacity, the growth rate starts to decline exponentially. As resources become limited, competition for food, space, and other necessities increases, leading to a decrease in the population growth rate.
For instance, imagine a population of rabbits in a forest. Initially, the population grows rapidly due to abundant resources. However, as the number of rabbits increases, the availability of food and space becomes limited. This leads to increased competition and a decrease in the population growth rate. The rate of decrease follows an exponential decay pattern.
To solidify your understanding of exponential decay, here's a practice problem for you:
Problem:
A radioactive substance has a half-life of 5 years. If the initial amount of the substance is 100 grams, how much will remain after 15 years?
Solution:
Since the half-life is 5 years, we can determine the number of half-lives that have occurred in 15 years by dividing 15 by 5. This gives us 3 half-lives.
After each half-life, the amount of the substance is reduced by half. So, after 3 half-lives, the amount remaining will be (1/2) * (1/2) * (1/2) = 1/8 of the initial amount.
Therefore, the amount remaining after 15 years is 1/8 * 100 grams = 12.5 grams.
I hope this explanation and practice problem help you understand the concept of exponential decay better.