Problem 10 - 5 Steps to A 5: AP Calculus AB 2021

Question

If the half-life of a radioactive element is 4500 years, and initially, there are 100 grams of this element, approximately how many grams are left after 5000 years?

Step-by-Step Solution

Answer

Answer: Approximately 46.32 grams.

Step 1: Write the formula for exponential decay

We will use the exponential decay formula to find the amount of radioactive element remaining after a certain time: \(A(t) = A_0(1/2)^{t/h}\)

Step 2: Plug in the given values

Now, we will plug in the given values of \(A_0\), \(t\), and \(h\) into the formula: \(A(t) = 100(1/2)^{5000/4500}\)

Step 3: Evaluate the expression

Simplify the expression and compute the remaining amount of the radioactive element: \(A(t) = 100(1/2)^{1.1111} \approx 46.32\) grams So, approximately 46.32 grams of the radioactive element are left after 5000 years.

Key Concepts

Half-Life

The concept of half-life is crucial in understanding radioactive decay. Half-life refers to the time it takes for half of a radioactive substance to decay. Every radioactive material has its unique half-life, which can vary from fractions of a second to billions of years.
For example, if you start with 100 grams of a radioactive element with a half-life of 4500 years, after 4500 years, only 50 grams will remain un-decayed. This process continues, so after 9000 years, you’d have 25 grams left.
Understanding half-life helps scientists predict how long it takes for a radioactive element to decrease to a certain level, which is important in fields like archeology, medicine, and nuclear energy.

Key Points about Half-Life:

• Specific to each radioactive element.
• Helps to predict the decay rate.
• Used in dating ancient artifacts.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy. This happens because the nucleus emits radiation in the form of particles or electromagnetic waves.
This emission changes the nucleus into a more stable configuration, eventually resulting in a stable isotope or element.
There are different types of radioactive decay, such as alpha decay, beta decay, and gamma decay. Each type involves different particles and radiation, and the decay type depends on the balance of forces within the nucleus.

Effects of Radioactive Decay:

• Changes the element into a different isotope.
• Involves the release of radiation.
• Used in carbon dating and medical treatments.

Exponential Functions

Exponential functions are a key concept in modeling situations where a quantity changes at a rate proportional to its current value. This makes them perfect for describing radioactive decay.
The basic form of an exponential decay function is \(A(t) = A_0 e^{-kt}\), where:

• \(A(t)\) is the amount remaining at time \(t\).
• \(A_0\) is the initial amount.
• \(k\) is a positive constant related to the decay rate.

For radioactive decay involving half-life, the formula becomes \(A(t) = A_0(1/2)^{t/h}\), which explains how the element's mass decreases over time.

Characteristics of Exponential Functions:

• Shows rapid decay initially, slowing over time.
• Used in populations, finance, and decay processes.
• Easy to manipulate with logarithms for calculations.

Understanding these functions allows you to predict how much of a substance remains after a given period, which is crucial in labs and fields managing radioactive materials.