In 4 years, 30% of a radioactive element decays. Find its half-life. (Round your answer to one decimal place.)
_____ yr
(1/2)^(4/k) = 0.7
Now just solve for k.
Expect a number greater than 4, since if the half-life were 4, then 50% would decay in 4 years.
In other words
(1/2)^([Years]/X)=[% remaining]
Solve for X
To find the half-life of a radioactive element, we can use the formula:
t(1/2) = (ln(2))/k
Where t(1/2) is the half-life, ln(2) is the natural logarithm of 2 (approximately 0.693), and k is the decay constant.
In this case, we know that after 4 years, 30% of the element decays. This means that 70% of the original amount remains.
To find the decay constant, we can use the formula:
k = -(ln(remaining amount)/time)
Substituting the given values:
k = -(ln(0.70)/ 4)
Calculating this:
k ≈ -0.112
Finally, we can substitute this decay constant into the half-life formula:
t(1/2) = (ln(2))/k
t(1/2) = 0.693 / 0.112
Calculating this:
t(1/2) ≈ 6.18 years
Therefore, the half-life of the radioactive element is approximately 6.18 years.
To find the half-life of a radioactive element, we need to determine the time it takes for half of the substance to decay.
In this case, we know that in 4 years, 30% of the element decays. Therefore, after 4 years, we are left with 70% of the original substance.
To find the time it takes for the substance to decay by half, we can set up the following equation:
(Original substance) * (decay rate)^(time period) = (Remaining substance)
Let's assign the original substance a value of 100.
Then, we have:
100 * (0.70)^(time period) = 50
Dividing both sides by 100, we get:
(0.70)^(time period) = 0.5
To solve for the time period (half-life), we can take the logarithm (base 0.70) of both sides:
log base 0.70 (0.70)^(time period) = log base 0.70 (0.5)
Since the logarithm of a number to its own base is always equal to 1, the left side simplifies to:
time period = log base 0.70 (0.5)
Using a calculator or a math software, we find that the time period is approximately 2.3 years. Therefore, the half-life of the radioactive element is approximately 2.3 years.
Rounding to one decimal place, the answer is:
2.3 yr