Problem 31 - Algebra for College Students

Question

Strontium- 90 has a half-life of 29 years. If there are 400 grams of strontium- 90 initially, how much, to the nearest gram, will remain after 87 years? After 100 years? \(\quad 50\) grams; 37 grams

Step-by-Step Solution

Answer

50 grams remain after 87 years, and 37 grams after 100 years.

Step 1: Understand the half-life concept

Half-life is the time required for half of the radioactive substance to decay. For Strontium-90, the half-life is 29 years.

Step 2: Write the decay formula

The amount of a radioactive substance remaining after time \( t \) can be calculated using the formula: \[ A(t) = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} \]where \( A(t) \) is the remaining amount, \( A_0 \) is the initial amount, and \( T \) is the half-life.

Step 3: Calculate strontium-90 remaining after 87 years

Plug the values into the formula: \[ A(87) = 400 \times \left(\frac{1}{2}\right)^{\frac{87}{29}} \]This simplifies to \[ A(87) = 400 \times \left(\frac{1}{2}\right)^3 = 400 \times \frac{1}{8} = 50 \] So, around 50 grams will remain after 87 years.

Step 4: Calculate strontium-90 remaining after 100 years

Again, use the decay formula for \( t = 100 \): \[ A(100) = 400 \times \left(\frac{1}{2}\right)^{\frac{100}{29}} \]This simplifies to \[ A(100) \approx 400 \times 0.092 = 36.8 \] Rounding to the nearest gram, around 37 grams will remain after 100 years.

Step 5: Verify calculations for accuracy

Verify the calculations by checking the simplifications of powers and multiplications to ensure the steps have no errors.

Key Concepts

Radioactive Decay

Radioactive decay is a fascinating natural process. It refers to the way unstable atomic nuclei lose energy by emitting radiation. This process changes the atom into a different element or isotope. Radiation can be in the form of alpha particles, beta particles, or gamma rays.
Strontium-90, for example, breaks down over time through beta decay, creating Yttrium-90.
Understanding radioactive decay is crucial in different fields like archaeology, medicine, and nuclear energy. It helps in carbon dating of fossils, measuring sterilization of medical equipment, and managing nuclear waste. One of the most important aspects here is the concept of half-life, which helps us determine how long it takes for half of the radioactive atoms in a sample to decay.

Exponential Decay Formula

The exponential decay formula is a mathematical expression that helps calculate how much of a substance remains after a certain period.
If you are dealing with radioactive substances like Strontium-90, this formula becomes invaluable. The formula is given by: \[A(t) = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}\] Here:

• \( A(t) \) is the amount remaining after time \( t \).
• \( A_0 \) is the initial quantity of the substance.
• \( T \) is the half-life of the substance.

Using this formula allows you to estimate how much of a radioactive substance remains over specified periods. Calculating how much Strontium-90 remains after diving its lifespan and decay can illustrate the practical applications of this formula. This formula is not just limited to physics; it can be used to model different phenomena, like cooling rates and population declines.

Strontium-90 Decay

Strontium-90 is a byproduct of nuclear reactions, often found in nuclear fallout. With a half-life of 29 years, this radioactive isotope gradually decays, posing long-term environmental risks. Understanding Strontium-90 decay can aid in safeguarding ecosystems.
In practical terms, consider starting with 400 grams of Strontium-90. After 29 years, half will decay, leaving 200 grams. After another 29 years, half of the remaining 200 grams will decay, leaving you with 100 grams.
After tackling this initial problem, the calculated amount of Strontium-90 remaining aids in decision making for safe storage or containment. Applications for this computation span across various areas, such as environmental science and nuclear safety protocols, emphasizing the importance of accurate mathematical analysis in real-world scenarios.

Mathematical Problem Solving

Mathematical problem solving is a powerful tool. It involves interpreting real-world situations through the lens of mathematics. In the context of radioactive decay, you engage with mathematical formulas to arrive at an understanding of how substances change over time.
When given a problem like determining Strontium-90 decay over several decades, breaking it into manageable steps becomes key. First, clearly identify the problem and relevant formulas. Consistently verify each computational step.
Applying the decay formula involves substituting known values into algebraic equations. This demand for accuracy considers every variable: initial quantity, time periods, and the substance's half-life.
Always double-check calculations for precision. Estimating results aids in intuition, while verifying through calculations ensures accuracy in numerical representations. By doing so, you not only solve the problem at hand but also build confidence and skill in mathematical reasoning.