A. Concept of "Decay by a Constant Percent Rate Per Unit Interval"
Definition of Key Terms:
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Decay: A process in which a quantity decreases over time. In this context, decay refers to the reduction in quantity at a specified rate.
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Constant Percent Rate: This implies that the quantity decreases by the same percentage during each time interval. For example, if something decays at a rate of 10%, it loses 10% of its current value in each time cycle.
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Unit Interval: The defined period over which the decay occurs. This could be one second, one hour, one day, etc., depending on the context of the scenario being studied.
Example: Suppose a radioactive substance has a decay rate of 5% per week. If we start with an initial amount of 100 grams, after the first week, the amount will be:
- Week 1: \(100 \times (1 - 0.05) = 100 \times 0.95 = 95\) grams.
- Week 2: \(95 \times (1 - 0.05) = 95 \times 0.95 = 90.25\) grams.
This process continues, with the substance retaining 95% of its quantity each successive week.
B. Real-World Scenario of Constant Percent Rate Decay
Scenario: Radioactive Material Decay
Radioactive materials decay over time at a constant percent rate. For instance, Carbon-14 isotopes have a half-life of approximately 5730 years, meaning that every 5730 years, half of the remaining Carbon-14 decays.
Significance: Understanding radioactive decay is critical in fields such as archaeology (carbon dating), nuclear medicine, radiation safety, and environmental science. By studying the decay rate, scientists can determine the age of ancient artifacts or assess the impact of radiation on living organisms, enabling informed decisions about health and safety.
C. Exponential Decay Equation
Equation: \[ N(t) = N_0 \times (1 - r)^t \]
Where:
- \(N(t)\) = the remaining quantity at time \(t\).
- \(N_0\) = the initial quantity of the substance (e.g., grams of a radioactive element).
- \(r\) = decay rate (expressed as a decimal, e.g., 5% = 0.05).
- \(t\) = the number of time intervals.
Example: For Carbon-14, if \(N_0 = 100\) grams and \(r = 0.05\) (5%), the equation would be: \[ N(t) = 100 \times (1 - 0.05)^t = 100 \times 0.95^t \]
D. Calculation of Remaining Quantity
Using the Equation from Part C, calculate the remaining quantity after various time intervals:
| Time Intervals (\(t\)) | Remaining Quantity (\(N(t)\)) | |-------------------------|----------------------------------| | 0 | \(100 \times 0.95^0 = 100\) | | 1 | \(100 \times 0.95^1 = 95\) | | 2 | \(100 \times 0.95^2 \approx 90.25\) | | 3 | \(100 \times 0.95^3 \approx 85.74\) | | 4 | \(100 \times 0.95^4 \approx 81.24\) | | 5 | \(100 \times 0.95^5 \approx 77.14\) |
Explanation: To find \(N(t)\) for each \(t\), we substitute the value of \(t\) into the equation \(N(t) = 100 \times 0.95^t\).
E. Graphing the Relationships
To graph the relationships between time and the remaining quantity, we plot the values from the table created in Part D.
- X-axis: Time intervals \(t\) (0 to 5)
- Y-axis: Remaining Quantity \(N(t)\)
As \(t\) increases, the quantity \(N(t)\) will show an exponential decay pattern, typically resembling a downward curve.
Graph Representation (Pseudocode):
- Create a line graph using the points gathered:
import matplotlib.pyplot as plt
# Data for graphing
t_values = [0, 1, 2, 3, 4, 5] # Time intervals
N_values = [100, 95, 90.25, 85.74, 81.24, 77.14] # Remaining quantities
# Plotting
plt.plot(t_values, N_values, marker='o')
plt.title("Exponential Decay of Radioactive Material")
plt.xlabel("Time Intervals (t)")
plt.ylabel("Remaining Quantity (grams)")
plt.grid()
plt.show()
This code creates a visual representation, indicating how the quantity decays over time, reinforcing the understanding of the exponential nature of decay by a constant percent rate.