Use the exponential decay model A = The half-life of thorium-229 is 7310 years. What is the decay rate k for thorium-229? (round six decimal places) k =-0.00009 0 How long will it take for a sample of this substance to decay to 20 percent of its original amount? (round one decimal place) It will take 16973.3 x years. Show the following steps in your work. Apekt to solve the following. Show the differential equation. Show the work to separate the equation and integration to find the general solution. Show the substitution of the intital value to find the particular solution.
Use the exponential decay model A = The half-life of thorium-229 is 7310 years. What is the decay rate k for thorium-229? (round six decimal places) k =-0.00009 0 How long will it take for a sample of this substance to decay to 20 percent of its original amount? (round one decimal place) It will take 16973.3 x years. Show the following steps in your work. Apekt to solve the following. Show the differential equation. Show the work to separate the equation and integration to find the general solution. Show the substitution of the intital value to find the particular solution.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Exponential Decay Model: Thorium-229
To solve problems related to the decay of Thorium-229, we use the exponential decay model given by:
\[ A = A_0 e^{kt} \]
#### Given Information:
- **Half-life of Thorium-229:** 7310 years
#### Problem 1: Calculate the Decay Rate \( k \)
**Question:** What is the decay rate \( k \) for Thorium-229? (Round to six decimal places)
**Answer:**
\[ k = -0.00009 \]
#### Problem 2: Time for Decay to 20% of Original Amount
**Question:** How long will it take for a sample of this substance to decay to 20 percent of its original amount? (Round to one decimal place)
**Answer:**
\[ \text{It will take } 16973.3 \text{ years.} \]
#### Required Steps to Solve:
1. **Show the Differential Equation:**
- Start with the differential equation governing exponential decay.
2. **Separate the Equation and Integrate:**
- Demonstrate the separation of variables and perform the integration necessary to find the general solution.
3. **Substitute the Initial Value:**
- Use the given initial conditions (half-life) to solve for the specific decay rate and time required for decay to a certain percentage.
By following these steps, you can accurately model the decay process using the provided exponential decay formula.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4c907f3-6692-4a58-8d85-7b416968270c%2F5d35899c-a86f-4c91-bdf0-fb4f4a95694a%2Fxd9ifto_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exponential Decay Model: Thorium-229
To solve problems related to the decay of Thorium-229, we use the exponential decay model given by:
\[ A = A_0 e^{kt} \]
#### Given Information:
- **Half-life of Thorium-229:** 7310 years
#### Problem 1: Calculate the Decay Rate \( k \)
**Question:** What is the decay rate \( k \) for Thorium-229? (Round to six decimal places)
**Answer:**
\[ k = -0.00009 \]
#### Problem 2: Time for Decay to 20% of Original Amount
**Question:** How long will it take for a sample of this substance to decay to 20 percent of its original amount? (Round to one decimal place)
**Answer:**
\[ \text{It will take } 16973.3 \text{ years.} \]
#### Required Steps to Solve:
1. **Show the Differential Equation:**
- Start with the differential equation governing exponential decay.
2. **Separate the Equation and Integrate:**
- Demonstrate the separation of variables and perform the integration necessary to find the general solution.
3. **Substitute the Initial Value:**
- Use the given initial conditions (half-life) to solve for the specific decay rate and time required for decay to a certain percentage.
By following these steps, you can accurately model the decay process using the provided exponential decay formula.
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