https://file.aiquickdraw.com/mathbot/file/1727113228171owby99nv.jpg

The problem provides a radioactive decay model for Samarium-151 (Sm-151) using the differential equation:

\[
\frac{dy}{dt} = -0.0077y
\]

This is a first-order linear differential equation describing exponential decay. We are asked to find the half-life of Sm-151.

### Step-by-step solution:

1. **General Form of Exponential Decay**:  
   The general solution to the differential equation \(\frac{dy}{dt} = -ky\) is given by:
   \[
   y(t) = y_0 e^{-kt}
   \]
   where \(y_0\) is the initial quantity, \(k\) is the decay constant, and \(t\) is time.

2. **Decay Constant**:  
   From the differential equation, we can see that \(k = 0.0077\) per year.

3. **Half-life Formula**:  
   The half-life (\(t_{\frac{1}{2}}\)) is the time it takes for the substance to decay to half its initial value. Using the half-life formula for exponential decay:
   \[
   t_{\frac{1}{2}} = \frac{\ln(2)}{k}
   \]
   
4. **Calculate the Half-life**:  
   Substituting \(k = 0.0077\) into the formula:
   \[
   t_{\frac{1}{2}} = \frac{\ln(2)}{0.0077}
   \]

Let's calculate this value.The half-life of Samarium-151 (Sm-151) is approximately **90.02 years**.

Would you like any further details or have any questions?

Here are five related questions for deeper understanding:
1. How does the decay constant \(k\) relate to the speed of radioactive decay?
2. What is the significance of the half-life in radioactive decay problems?
3. Can the half-life change for different isotopes of the same element?
4. How would you model a scenario where multiple substances are decaying simultaneously?
5. What happens mathematically to the amount of Sm-151 after a very long time?

**Tip:** Remember, exponential decay models are commonly used in fields like physics, chemistry, and finance to describe processes where quantities reduce over time at a rate proportional to their current value.

The radioactive decay of Sm-151 (an isotope of samarium) can be modeled by the differential equation dy/dt = -0.0077y, where t is measured in years. Find the half-life of Sm-151.

Learn how to calculate the half-life of Samarium-151 (Sm-151) using a first-order linear differential equation dy/dt = -0.0077y. This detailed step-by-step solution covers the exponential decay model and half-life formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation