Radioactive Decay: Calculating Remaining Mass Over Time
Hey guys! Ever wondered how scientists figure out how much of a radioactive substance is left after a certain amount of time? It's all about understanding something called radioactive decay. Let's break down a cool math problem related to this. We'll cover how to find a function that shows how much of a radioactive substance remains over time and then use that to calculate the remaining amount after a specific period. Buckle up, because we're diving into some interesting stuff!
Understanding the Problem: Radioactive Decay and Half-Life
Okay, so here's the scenario: We have a radioactive substance weighing 320 grams. This particular substance has a half-life of 10 months. Now, what exactly does 'half-life' mean? Simple! It's the time it takes for half of the substance to decay or transform into something else. Think of it like this: after 10 months, our 320-gram sample will become 160 grams. After another 10 months (a total of 20 months), half of that 160 grams will decay, leaving us with 80 grams, and so on. This process continues indefinitely, with the substance gradually decreasing in mass. Our mission, should we choose to accept it, is to figure out a formula that lets us predict how much of this substance will be left after any given amount of time. Then, we will apply the formula to determine how much will be left after 30 months.
This problem combines concepts from exponential decay, which is a mathematical model that describes how a quantity decreases over time, and the specifics of radioactive decay. Radioactive decay is a naturally occurring process where unstable atomic nuclei lose energy by emitting radiation. The rate of decay is constant, which is what allows us to use the half-life to make accurate predictions about the remaining mass. So, the half-life is a fundamental characteristic of each radioactive isotope, and it is the key to unlocking the secrets of its decay process. Understanding this concept is critical to solving our problem. The goal is to craft a function that maps the time elapsed to the remaining mass of the radioactive substance. This is achieved by using the concept of exponential decay and the given half-life of the substance. Let's look deeper into how to build our formula.
(a) Finding the Function for Remaining Radioactive Substance
Alright, time to put on our thinking caps! To find the function, we need to understand that the mass of the substance decreases exponentially. This means the decrease isn't linear; it's not like we're taking away the same amount each month. Instead, the mass is halved every 10 months. We can use the general form of an exponential decay function: M(t) = M₀ * (1/2)^(t/T).
Where:
M(t)is the mass of the substance remaining after timet(in months).M₀is the initial mass of the substance (320 grams in our case).tis the time elapsed (in months).Tis the half-life of the substance (10 months).
Let's plug in the values we know:
M(t) = 320 * (1/2)^(t/10)
This, my friends, is our function! It tells us the mass of the radioactive substance remaining after 't' months. Now, let's decode this equation: The initial mass M₀ is multiplied by (1/2) raised to the power of t divided by the half-life T. The (1/2) part is crucial since the substance is halved with each half-life. The term t/10 determines how many half-lives have passed during the given time t. This means the mass is halved after every 10-month interval. The equation elegantly captures the essence of exponential decay, making it possible to predict the future mass of our radioactive substance. The result is a powerful tool that allows us to analyze and comprehend the decay of radioactive materials over time, reflecting the changes in the amount of the substance. This is an elegant, compact representation of the exponential decay process, which simplifies our understanding and calculations of the remaining mass of radioactive substances over time. Understanding how to derive and use the exponential decay function is essential in this type of problem.
(b) Calculating the Remaining Substance After 30 Months
Now that we have our function, let's calculate how much of the substance is left after 30 months. We'll use the function we just found: M(t) = 320 * (1/2)^(t/10). Let's substitute t = 30 months.
M(30) = 320 * (1/2)^(30/10)
M(30) = 320 * (1/2)³
M(30) = 320 * (1/8)
M(30) = 40
So, after 30 months, there will be 40 grams of the radioactive substance remaining. After 30 months, the initial amount has gone through three half-lives. The initial 320 grams are first reduced to 160 grams after the first 10 months. After the second 10 months, the remaining mass is 80 grams. After the third 10 months, the remaining mass is 40 grams. To recap, after 30 months the substance went through 3 half-lives, decreasing the initial mass to a quarter, then halved it one more time, and the initial amount decreased from 320 grams to 40 grams. This calculation highlights the practical application of the exponential decay function and how half-life influences the quantity of radioactive materials over time. Understanding these types of calculations is important, because they help us understand the behavior of radioactive materials and other things that decay over time.
To sum up, the problem demonstrated how to formulate an exponential decay function that models the reduction in mass over time, a core concept in mathematics and physics. The initial mass and the half-life were essential in determining the function and calculating the remaining mass. Radioactive decay is a naturally occurring phenomenon and understanding its dynamics is vital. The ability to calculate the remaining mass after a certain period of time is essential. The exponential decay model helps to comprehend the nature of radioactive substances. This problem is an example of how math can be used to model real-world problems and is a good example of a math problem in real life.
