This exercise uses the radioactive decay model. The half-life of radium-226 is 1600 years. Suppose we have a 29-mg sample. (a) Find a function m(t) = mo2-t/h that models the mass remaining after t years. m(t) = 29 2 (b) Find a function m(t) = moe-rt that models the mass remaining after t years. (Round your r value to six decimal places m(t) = (c) How much of the sample will remain after 5000 years? (Round your answer to one decimal place.) X mg (d) After how many years will only 17 mg of the sample remain? (Round your answer to one decimal place.) X yr
Radioactive decay
The emission of energy to produce ionizing radiation is known as radioactive decay. Alpha, beta particles, and gamma rays are examples of ionizing radiation that could be released. Radioactive decay happens in radionuclides, which are imbalanced atoms. This periodic table's elements come in a variety of shapes and sizes. Several of these kinds are stable like nitrogen-14, hydrogen-2, and potassium-40, whereas others are not like uranium-238. In nature, one of the most stable phases of an element is usually the most prevalent. Every element, meanwhile, has an unstable state. Unstable variants are radioactive and release ionizing radiation. Certain elements, including uranium, have no stable forms and are constantly radioactive. Radionuclides are elements that release ionizing radiation.
Artificial Radioactivity
The radioactivity can be simply referred to as particle emission from nuclei due to the nuclear instability. There are different types of radiation such as alpha, beta and gamma radiation. Along with these there are different types of decay as well.
![# Radioactive Decay Model
This exercise demonstrates the application of a radioactive decay model.
### Problem Statement:
The half-life of radium-226 is 1600 years. Suppose we have a 29-mg sample.
### Questions:
#### (a) Find a function \( m(t) = m_0 2^{-t/h} \) that models the mass remaining after \( t \) years.
Given function to find:
\[ m(t) = 29 \cdot 2^{-\left( \frac{t}{1600} \right)} \]
The answer to this part is correct, as indicated by the green check mark.
#### (b) Find a function \( m(t) = m_0 e^{-rt} \) that models the mass remaining after \( t \) years. (Round your \( r \) value to six decimal places.)
This part requires finding the decay constant \( r \) and subsequently expressing the function with the exponential model. However, the answer entered was incorrect, marked by the red cross.
#### (c) How much of the sample will remain after 5000 years? (Round your answer to one decimal place.)
Solution box:
\[ \boxed{} \text{ mg} \]
This part also appears to be answered incorrectly, as indicated by the red cross.
#### (d) After how many years will only 17 mg of the sample remain? (Round your answer to one decimal place.)
Solution box:
\[ \boxed{} \text{ yr} \]
This part was also marked incorrect, as evidenced by the red cross.
### Visual Elements:
The image includes the following visual components:
- A properly completed equation box in part (a) with a green check mark indicating a correct response.
- Empty solution boxes for parts (b), (c), and (d) with red crosses indicating incorrect answers.
### Instructions:
For solving parts (b), (c), and (d):
- Calculate the decay constant \( r \) using the relationship between half-life and the exponential model.
- Use the derived function to determine the amount of sample remaining after a specified number of years or the time taken for the sample to decay to a specified amount.
With correct mathematical steps and understanding, complete the given radioactive decay model exercise.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5451aef3-1a55-4404-bb01-8f8933157334%2F51a83ac0-2e7d-4c4a-a52d-5e34ee9141db%2Fuqaepsi_processed.jpeg&w=3840&q=75)
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