Problem 19 - Algebra A Combined Function

Question

Solve. Unless noted otherwise, round answers to the nearest whole. A rare isotope of a nuclear material is very unstable, decaying at a rate of \(15 \%\) each second. Find how much isotope remains 10 seconds after 5 grams of the isotope is created.

Step-by-Step Solution

Answer

Approximately 1 gram of the isotope remains after 10 seconds.

Step 1: Understanding the Decay Formula

The decay formula follows exponential decay, expressed as \( A(t) = A_0 \times (1 - r)^t \), where \( A(t) \) is the amount of substance remaining after time \( t \), \( A_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is the time elapsed.

Step 2: Identifying Given Values

We are given the initial amount \( A_0 = 5 \) grams, the decay rate \( r = 0.15 \) (which is 15% per second), and the time \( t = 10 \) seconds. These values will be substituted into the decay formula.

Step 3: Substituting Values into the Formula

Substitute \( A_0 = 5 \), \( r = 0.15 \), and \( t = 10 \) into the decay formula: \[A(10) = 5 \times (1 - 0.15)^{10}\]

Step 4: Simplifying the Equation

First, calculate \( 1 - 0.15 = 0.85 \). Then substitute this back into the equation, so it becomes: \[A(10) = 5 \times 0.85^{10}\]

Step 5: Calculating the Remaining Isotope

Now calculate \( 0.85^{10} \), which is approximately \( 0.1969 \). Then multiply by 5 grams: \[A(10) = 5 \times 0.1969 \approx 0.9845\]

Step 6: Rounding to the Nearest Whole

Round \( 0.9845 \) to the nearest whole number, which is 1 gram. So, approximately 1 gram of the isotope remains after 10 seconds.

Key Concepts

Decay Formula

Exponential decay is a process in which a quantity decreases at a rate proportional to its current value. The decay formula used in our problem is - \( A(t) = A_0 \times (1 - r)^t \).This formula allows us to calculate the remaining amount of a substance after a certain period of time has passed.

• \( A(t) \) - the remaining amount after time \( t \).
• \( A_0 \) - the initial amount of the substance.
• \( r \) - the decay rate, expressed as a decimal.
• \( t \) - the time that has elapsed.

In our example, the decay rate is \( 15\% \) per second. After 10 seconds, you would substitute into the equation: - \( A(10) = 5 \times (1 - 0.15)^{10} \). Finish by solving for \( A(10) \). This will give the amount of isotope remaining.

Isotope Decay

Isotopes are different forms of the same element, varying in the number of neutrons. Some isotopes are unstable and tend to lose energy by emitting radiation, a process called radioactive decay. This decay can be rapid, with some isotopes disappearing quickly, as seen in our example with the rare isotope decaying at \( 15\% \) per second.

To analyze isotope decay, we apply the exponential decay formula. It describes how the unstable isotope decreases over time. By understanding each component:- The initial quantity \( A_0 \) is the starting amount of the isotope.- The decay rate \( r \) tells us what portion is lost each second.

Considering the half-life can often be useful, which is the time required for half of the isotope to decay. In the problem solved, we calculated that after a few seconds, almost all the isotope decays, resulting in approximately 1 gram left. The decay calculations referenced indicate the rapid decline typical of unstable isotopes.

Nuclear Decay Rate

The nuclear decay rate is a critical factor in understanding how quickly a radioactive substance diminishes. It's the speed at which unstable isotopes transform into more stable elements. This rate is measured as the fraction of the substance reduced over a specific time.

For instance, if an isotope decays at a rate of \( 15\% \) per second, over time, this leads to a substantial decrease in its mass. In the exercise example, initially beginning with 5 grams, after 10 seconds of nuclear decay at this rate, only a small quantity remains.

• The decay rate \( r \) is crucial for predictive calculations.
• Understanding the rate helps determine how often the isotope emits radiation.
• This knowledge is invaluable in fields like medicine, archaeology, and energy production, where precise control and predictions regarding radioactive materials are vital.

Grasping the nuclear decay rate concept is important as it underpins the nature and timescales of radioactive decay across various applications.