Final answer:
The half-life of the radioactive element in question is approximately 65.66 days, calculated using the half-life formula and the given data of weight reduction from 71 to 54 grams over the span of 46 days.
Step-by-step explanation:
The student is asking for the half-life of a radioactive element that was found in an illegal waste dump. The sample started with 71 grams and after 46 days, only 54 grams were left. To find the half-life of this element, we can use the concept that after each half-life, half of the original amount of radioactive material remains.
Let's use the formula:
N = N0 * (1/2)^(t/T)
Where:
- N is the remaining amount of substance after time t,
- N0 is the original amount of substance,
- T is the half-life of the substance,
- t is the time elapsed.
We have N = 54 grams, N0 = 71 grams, and t = 46 days. We need to find T.
54 = 71 * (1/2)^(46/T)
To isolate T, we must use logarithms. Taking the natural log (ln) of both sides:
ln(54) = ln(71) + (46/T) * ln(1/2)
Solving for T gives us:
T = -46 / [(ln(54) - ln(71)) / ln(1/2)]
After calculating, we round to two decimal places:
T ≈ 65.66 days
So, the half-life of this radioactive element is approximately 65.66 days.