Problem 23 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If a radioactive element has a half-life of 1 minute, and if a container holds 32 g of the element at 1: 00 P.M., then the amount remaining at 1: 05 P.M. will be \(1 \mathrm{g} .\)

Step-by-Step Solution

Answer

True, 1 gram remains after 5 minutes, which matches the statement's claim.

Step 1: Understand Half-Life Concept

The half-life of a radioactive element is the time it takes for half of the substance to decay. If the half-life is 1 minute, then every minute, the substance's quantity reduces by half.

Step 2: Determine the Number of Half-Lives

Calculate how many half-lives occur between 1:00 P.M. and 1:05 P.M. Since each half-life is 1 minute, there are 5 minutes in this period, which means there are 5 half-lives.

Step 3: Calculate the Decrease in Quantity

Start with 32 grams and apply the half-life calculation repeatedly: \(\text{After 1 minute: } 32/2 = 16 \text{ grams}\), \(\text{After 2 minutes: } 16/2 = 8 \text{ grams}\), \(\text{After 3 minutes: } 8/2 = 4 \text{ grams}\), \(\text{After 4 minutes: } 4/2 = 2 \text{ grams}\), and \(\text{After 5 minutes: } 2/2 = 1 \text{ gram}\).

Step 4: Conclusion

After calculating, we find that exactly 1 gram remains after 5 minutes. Therefore, the statement is true.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting particles or radiation. This process transforms the living nucleus into a more stable one, releasing energy in the form of radiation. The element undergoing this decay is known as a radioactive element. A key feature of radioactive decay is its randomness, meaning it is impossible to predict which nucleus will decay at a given moment.
However, even though the decay of individual nuclei is random, large numbers of decaying atoms follow a predictable pattern over time. This characteristic is essential in helping scientists understand how the quantity of a radioactive substance evolves.

• This property forms the basis of measuring half-lives, which quantify the decay rate.
• In our example, every minute, half of the substance decays, reducing the original quantity significantly over a short time frame.
• This concept is crucial in fields such as nuclear physics, archaeology, and medicine.

Exponential Decay

Exponential decay describes processes that reduce a quantity by a consistent percentage over time. In radioactive decay, an exponential decay function portrays how the amount of a radioactive substance decreases over time. The base of key exponential functions is typically less than one, making each subsequent value smaller.For any radioactive element with a known half-life, we can model its decay using exponential functions. For example, a substance with a half-life of 1 minute will systematically reduce to half its size every minute. This factor is repeated, leading to a swift decrease in mass.

• The decay can be described using the equation \( N(t) = N_0 (1/2)^{t/T} \), where \( N(t) \) is the remaining quantity at time \( t \), \( N_0 \) is the initial quantity, and \( T \) represents the half-life.
• For our exercise, starting from 32 grams, after 5 minutes, only a small fraction, specifically 1 gram, remains.

This characteristic helps model real-world phenomena where decay rates are predictable and consistent.

Mathematical Reasoning

Mathematical reasoning involves using logic and organized steps to solve problems. In the case of radioactive decay, we employ this reasoning to understand how quantities of radioactive materials decrease over time. The process requires understanding the rules governing half-lives and applying them iteratively to determine the quantity remaining. In the example provided, calculating the remaining radioactive material involved.

• Identifying the given half-life (1 minute).
• Calculating the number of half-lives over the time interval (5 minutes = 5 half-lives).
• Applying the half-life equation systematically for each time increment.

To conclude that only 1 gram remains after 5 minutes, mathematical reasoning allowed for organized thought, ensuring the steps logically followed one another. Such reasoning is crucial for solving problems reliably and accurately.