After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1996 , the hay in that country was contaminated by a radioactive isotope (half-life 5 days). If it is safe to feed the hay to cows when 11% of the radioactive isotope remains, how long did the farmers need to wait to use this hay? The farmers needed to wait approximately days for it to be safe to feed the hay to the cows (Round to one decimal place as needed.)
To find out how long the farmers needed to wait, we can use the formula for radioactive decay:
N(t) = N0 * (1/2)^(t/h)
Where:
N(t) is the amount of radioactive isotope remaining at time t
N0 is the initial amount of radioactive isotope
t is the time that has passed
h is the half-life of the radioactive isotope
We know that N(t) = 0.11 and we want to find t.
0.11 = 1 * (1/2)^(t/5)
Taking the logarithm of both sides:
log(0.11) = log(1/2)^(t/5)
Using the logarithmic property log(a^b) = b * log(a):
log(0.11) = (t/5) * log(1/2)
Dividing both sides by log(1/2):
(t/5) = log(0.11) / log(1/2)
t = (5 * (log(0.11) / log(1/2)))
Calculating this expression:
t ≈ 14.12
Therefore, the farmers needed to wait approximately 14.1 days for it to be safe to feed the hay to the cows.
Bot, when I worked the problem I came up with 15.9. Is that correct?
Apologies for the error. Let's recalculate it:
t = (5 * (log(0.11) / log(1/2)))
t ≈ 15.9
Yes, you are correct. The farmers needed to wait approximately 15.9 days for it to be safe to feed the hay to the cows.
To determine how long the farmers needed to wait for the hay to be safe to feed to the cows, we need to calculate the number of half-lives it takes for 11% of the radioactive isotope to remain.
1. Start with 100% of the radioactive isotope.
2. Each half-life reduces the amount of the isotope by half.
3. Therefore, after one half-life, the amount of the isotope remaining is 50%.
4. After two half-lives, the amount remaining is 25%.
5. After three half-lives, the amount remaining is 12.5%.
6. After four half-lives, the amount remaining is 6.25%.
Since we want 11% of the isotope to remain, the farmers needed to wait approximately four half-lives for the hay to be safe to feed to the cows.
Since the half-life of the isotope is 5 days, the farmers needed to wait approximately 4 * 5 = 20 days for it to be safe to feed the hay to the cows.
Therefore, the farmers needed to wait approximately 20 days for the hay to be safe to feed to the cows.