If given the following scenario:
The half-life of Sodium-24 is 15 hours. If you start with 600 grams of sodium-24 how much would be left after 4 days?
Which of the following equations could you use to solve for the amount of grams left after 4 days?
[tex]$A=600\left(\frac{1}{2}\right)^{\frac{15}{4}}$[/tex]
[tex]$A=600\left(\frac{1}{2}\right)^4$[/tex]
[tex]$A=600\left(\frac{1}{2}\right)^{\frac{4}{15}}$[/tex]
[tex]$A=600\left(\frac{1}{2}\right)^{\frac{96}{15}}$[/tex]
Answer (2)
Convert 4 days to 96 hours.
Apply the half-life formula: A = A 0 ( 2 1 ) t 1/2 t .
Substitute A 0 = 600 , t = 96 , and t 1/2 = 15 into the formula.
The correct equation is: A = 600 ( 2 1 ) 15 96 .
Explanation
Understanding the Problem We are given a problem involving radioactive decay and half-life. We need to determine the correct equation to calculate the amount of Sodium-24 remaining after 4 days, given its half-life is 15 hours and the initial amount is 600 grams.
Converting Time Units First, we need to convert the time to the same units. Since the half-life is given in hours, we convert 4 days to hours: 4 days × 24 day hours = 96 hours
Stating the Half-Life Formula The half-life formula is: A = A 0 ( 2 1 ) t 1/2 t where:
A is the amount remaining,
A 0 is the initial amount,
t is the time elapsed,
t 1/2 is the half-life.
Substituting Values Now, we substitute the given values into the formula:
A 0 = 600 grams
t = 96 hours
t 1/2 = 15 hours The equation becomes: A = 600 ( 2 1 ) 15 96
Identifying the Correct Equation Comparing the derived equation with the given options, we find that the correct equation is: A = 600 ( 2 1 ) 15 96
Final Answer Therefore, the correct equation to solve for the amount of grams left after 4 days is: A = 600 ( 2 1 ) 15 96
Examples
Radioactive decay is used in various applications such as carbon dating in archeology, medical treatments like radiation therapy, and determining the age of rocks in geology. Understanding half-life helps scientists and doctors accurately measure and predict the amount of radioactive material remaining over time, ensuring safety and effectiveness in these applications. For instance, in carbon dating, the amount of carbon-14 remaining in a sample is used to estimate the age of ancient artifacts.
To find the amount of Sodium-24 remaining after 4 days, convert 4 days to 96 hours. Use the half-life formula A = 600 ( 2 1 ) 15 96 . The correct equation is: A = 600 ( 2 1 ) 15 96 .
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