A sample of radioactive material is obtained from a very old rock. The activity of the rock over a period of time is monitored, and lnA is plotted as a function of t such as in Figure (b). The slope of the line has a value of -6.1 ×10^−8y^−1. FInd the half-life in years.
A sample of radioactive material is obtained from a very old rock. The activity of the rock over a period of time is monitored, and lnA is plotted as a function of t such as in Figure (b). The slope of the line has a value of -6.1 ×10^−8y^−1. FInd the half-life in years.
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A sample of radioactive material is obtained from a very old rock. The activity of the rock over a period of time is monitored, and lnA is plotted as a function of t such as in Figure (b). The slope of the line has a value of -6.1 ×10^−8y^−1.
FInd the half-life in years.
![**Problem 6:** The equation for the activity of a radioactive substance as a function of time is given by
\[ A = A_0 e^{-\lambda t}, \]
and is plotted in Figure (a). If we take the natural log of both sides of this equation, the result is the following:
\[ \ln A = -\lambda t + \ln A_0. \]
Notice that our equation now has the form of an equation of a line, \( y = mx + b \), with \( y = \ln A \) and a slope of \(- \lambda \). This line is plotted in Figure (b).
**Figure Descriptions:**
- **Figure (a):**
- Graph of \( A = A_0 e^{-\lambda t} \).
- The y-axis represents \( A \) (activity), and the x-axis represents \( t \) (time).
- The curve shows an exponential decay, starting from \( A_0 \) and decreasing over time.
- **Figure (b):**
- Graph of \( \ln A = -\lambda t + \ln A_0 \).
- The y-axis represents \( \ln A \), and the x-axis represents \( t \).
- The line has a negative slope of \(-\lambda\), indicating a linear relationship when using the natural logarithm transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2bd9187e-598c-4d81-ae39-6b008cf31e1f%2F0bf43ec1-eb37-4a41-9e96-0c093d705668%2Fg6frdpq_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6:** The equation for the activity of a radioactive substance as a function of time is given by
\[ A = A_0 e^{-\lambda t}, \]
and is plotted in Figure (a). If we take the natural log of both sides of this equation, the result is the following:
\[ \ln A = -\lambda t + \ln A_0. \]
Notice that our equation now has the form of an equation of a line, \( y = mx + b \), with \( y = \ln A \) and a slope of \(- \lambda \). This line is plotted in Figure (b).
**Figure Descriptions:**
- **Figure (a):**
- Graph of \( A = A_0 e^{-\lambda t} \).
- The y-axis represents \( A \) (activity), and the x-axis represents \( t \) (time).
- The curve shows an exponential decay, starting from \( A_0 \) and decreasing over time.
- **Figure (b):**
- Graph of \( \ln A = -\lambda t + \ln A_0 \).
- The y-axis represents \( \ln A \), and the x-axis represents \( t \).
- The line has a negative slope of \(-\lambda\), indicating a linear relationship when using the natural logarithm transformation.
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