In a lab experiment, the decay of a radioactive isotope is being observed. At the beginning of the first day of the experiment the mass of the substance was 1300 grams and mass was decreasing by 14% per day. Determine the mass of the radioactive sample at the beginning of the 11th day of the experiment. Round to the nearest tenth (if necessary).
after x days, the amount left is
f(x) = 1300 * 0.86^x
so now just find f(10)
Well, it seems like this radioactive isotope is having a real meltdown! Let's do some calculations to find out the mass at the beginning of the 11th day.
Since the mass is decreasing by 14% per day, that means each day it's only keeping 86% of its previous mass. So, we can use the formula:
Final Mass = Initial Mass × (0.86)^n
where n is the number of days.
Plugging in the values, we get:
Final Mass = 1300 × (0.86)^10
Calculating that, we find the final mass to be approximately 407.5 grams.
So, at the beginning of the 11th day, the mass of the radioactive sample is around 407.5 grams. Don't worry, it's not disappearing completely, just getting a little lighter each day!
To determine the mass of the radioactive sample at the beginning of the 11th day, we need to calculate the mass after each day of decay.
On the first day, the mass decreased by 14%, so we have:
Mass after 1st day = 1300 grams - (0.14 * 1300 grams)
Mass after 1st day = 1300 grams - 182 grams
Mass after 1st day = 1118 grams
On the second day, we again decrease the mass by 14% from the previous day's mass:
Mass after 2nd day = 1118 grams - (0.14 * 1118 grams)
Mass after 2nd day = 1118 grams - 156.52 grams
Mass after 2nd day = 961.48 grams
We can continue this process for the remaining days, or simply use the formula to calculate the mass after the 11th day using the initial mass of 1300 grams:
Mass after 11th day = Initial mass * (1 - decay rate/100)^(number of days)
Mass after 11th day = 1300 grams * (1 - 14/100)^11
Mass after 11th day ≈ 463.7 grams
Therefore, the mass of the radioactive sample at the beginning of the 11th day of the experiment is approximately 463.7 grams.
To determine the mass of the radioactive sample at the beginning of the 11th day, we need to calculate how much mass is left after 10 days of decay.
To do this, we will use the formula for exponential decay:
Mass(t) = Mass(0) * (1 - Decay Rate)^t
Where:
- Mass(t) is the mass at time t
- Mass(0) is the initial mass
- Decay Rate is the percentage by which the mass is decreasing
- t is the time in days
Given:
- Mass(0) = 1300 grams
- Decay Rate = 14% = 0.14
- t = 10 days (since we want to find the mass at the beginning of the 11th day)
Now, we can substitute the values into the formula:
Mass(10) = 1300 * (1 - 0.14)^10
Calculating the decay:
Mass(10) = 1300 * (0.86)^10
Mass(10) ≈ 1300 * 0.194860868
Mass(10) ≈ 252.41 grams (rounded to the nearest tenth)
Therefore, the mass of the radioactive sample at the beginning of the 11th day of the experiment is approximately 252.4 grams.