Problem 16 - Precalculus with Calculus Previews

Question

Strontium-9o is a dangerous radioactive substance found in acid rain. As such it can make its way into the food chain by polluting the grass in a pasture on which milk cows graze. The halflife of strontium-9o is 29 years. (a) Find an exponential model (3) for the amount remaining after \(t\) years. (b) Suppose a pasture is found to contain Str-90 that is 3 times a safe level \(A_{0} .\) How long will it be before the pasture can be used again for grazing cows?

Step-by-Step Solution

Answer

(a) Model: \(A(t) = A_0 e^{-0.0239t}\). (b) Time: about 46 years.

Step 1: Understanding the Halflife Formula

The half-life of a substance is the time it takes for half of the substance to decay. For Strontium-90, the half-life is 29 years. The exponential decay model can be expressed as:\[A(t) = A_0 e^{-kt}\]where \(A(t)\) is the amount remaining at time \(t\), \(A_0\) is the initial amount, and \(k\) is the decay constant.

Step 2: Finding the Decay Constant

To find the decay constant \(k\), we use the half-life formula:\[k = \frac{\ln(2)}{T_{1/2}}\]For Strontium-90, \(T_{1/2} = 29\) years. Thus:\[k = \frac{\ln(2)}{29} \approx 0.0239\]

Step 3: Formulate the Exponential Model for Part (a)

Substitute \(k = 0.0239\) into the decay formula to get the exponential model:\[A(t) = A_0 e^{-0.0239t}\]This model describes the decay of Strontium-90 over time.

Step 4: Solve for Time Until Safe Level for Part (b)

Given that the pasture has an amount \(3A_0\) and needs to decay to \(A_0\), set up the equation:\[A(t) = A_0 = 3A_0 e^{-0.0239t}\]Simplify:\[1 = 3e^{-0.0239t}\]Divide by 3:\[\frac{1}{3} = e^{-0.0239t}\]

Step 5: Solving the Exponential Equation

Take the natural logarithm on both sides to solve for \(t\):\[\ln\left(\frac{1}{3}\right) = -0.0239t\]Rearrange and solve for \(t\):\[t = \frac{\ln\left(\frac{1}{3}\right)}{-0.0239} \approx 45.63 \text{ years}\]

Step 6: Conclusion

The exponential model for the decay of Strontium-90 is \(A(t) = A_0 e^{-0.0239t}\). It will take approximately 46 years for the Strontium-90 levels to decrease to a safe level in the pasture.

Key Concepts

Half-life

When discussing radioactive substances like Strontium-90, one of the key concepts is half-life. The half-life is defined as the time it takes for half of a given quantity of a radioactive substance to decay. This concept is crucial in understanding radioactive decay because it provides insights into how long a substance remains active.

For Strontium-90, the half-life is 29 years, meaning that every 29 years, the quantity of Strontium-90 will reduce by half. This gradual reduction over time is what necessitates careful monitoring, especially in environments where it's found, such as pastures or water sources.

Knowing the half-life helps to predict how long it will take before the radioactive element becomes less hazardous or reaches a level that is considered safe. In practical scenarios, half-life is used to determine the safe usage period of areas contaminated by radioactive substances.

Decay Constant

The decay constant is an important value in the exponential decay model, representing the rate at which a substance undergoes radioactive decay. It's denoted by the symbol \(k\) and is specific to each radioactive isotope.

To calculate the decay constant, scientists use the formula:

• \( k = \frac{\ln(2)}{T_{1/2}} \)

This formula utilizes the natural logarithm of 2 (because of the half-life definition) and the half-life \( T_{1/2} \) of the substance. For Strontium-90, we found \( k \approx 0.0239 \) per year.

The decay constant is crucial in setting up the exponential decay formula \( A(t) = A_0 e^{-kt} \), where it governs how quickly or slowly the substance decays. A higher decay constant would indicate a faster rate of decay.

Radioactive Decay

Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting radiation. This process transforms the original, unstable atom into a different atom, which may or may not be stable.

The exponential decay formula \( A(t) = A_0 e^{-kt} \) is a powerful tool for modeling this decay process. In this equation:

• \( A(t) \) is the amount of substance remaining after time \( t \).
• \( A_0 \) represents the initial amount of the substance.
• \( k \) is the decay constant specific to the substance.

This formula shows how the initial quantity of the substance reduces exponentially over time. Radioactive decay is fundamentally unpredictable on a particle level but statistically predictable, allowing people to make accurate forecasts about safety levels and timing.

Understanding radioactive decay helps us grasp not just the risks posed by contamination, but also the timelines involved in reducing those risks.