The half-life of an isotope known as colbalt-60 is 5.26 years. What is the decay factor of this substance? (Hint: use the equation [tex]$y=c a^x$[/tex])
A. 0.877
B. 2.000
C. 0.026
D. 1.140
Answer (1)
Recognize that after one half-life (5.26 years), the remaining amount is half the initial amount.
Substitute x = 5.26 and y = c /2 into the equation y = c a x , resulting in 2 1 = a 5.26 .
Solve for a by taking the 5.26th root: a = ( 2 1 ) 5.26 1 .
Calculate the value of a , which is approximately 0.877 .
Explanation
Understanding the Problem We are given that the half-life of Cobalt-60 is 5.26 years. This means that after 5.26 years, half of the initial amount of Cobalt-60 remains. We are also given the decay equation y = c a x , where y is the amount of substance remaining after time x , c is the initial amount, and a is the decay factor. Our goal is to find the decay factor a .
Setting up the Equation After one half-life, which is 5.26 years, the amount of substance remaining is half of the initial amount. So, we can set x = 5.26 and y = 2 c in the equation y = c a x .
Substituting Values Substituting these values into the equation, we get: 2 c = c a 5.26 Dividing both sides by c , we have: 2 1 = a 5.26 Now, we need to solve for a .
Solving for the Decay Factor To solve for a , we take the 5.26th root of both sides: a = ( 2 1 ) 5.26 1 Calculating this value, we find: a ≈ 0.8765 Rounding to three decimal places, we get a ≈ 0.877 .
Final Answer Therefore, the decay factor of Cobalt-60 is approximately 0.877.
Examples
Understanding radioactive decay is crucial in various fields, such as medicine and environmental science. For instance, Cobalt-60, with its known half-life and decay factor, is used in radiation therapy to treat cancer. By knowing the decay factor, medical professionals can accurately calculate the dosage required to target cancerous cells while minimizing harm to healthy tissue. Similarly, in environmental monitoring, understanding the decay of radioactive isotopes helps in assessing the long-term impact of nuclear accidents and managing radioactive waste.
