The decay of uranium is modelled by
D = D_0 * 2^(-kt). If it takes 6 years for the mass of uranium to halve, find the percentage remaining after:
a. 2 years
Since the half-life is 6 years,
D = D_0 * 2^(-t/6)
So, just plug in your numbers and find that
D/D_0 = 2^(-2/6) = 0.7937 ≈ 79%
Alright, thanks
To find the percentage remaining after 2 years, we first need to determine the decay constant, k.
Given that it takes 6 years for the mass of uranium to halve, we can use this information to form an equation:
D = D_0 * 2^(-kt)
where D is the remaining mass after time t, D_0 is the initial mass, and k is the decay constant.
After 6 years, the mass halves, which means D/D_0 = 1/2. We can substitute these values into the equation:
1/2 = 2^(-6k)
To solve for k, we can take the logarithm of both sides of the equation:
log(1/2) = log(2^(-6k))
-1 = -6k * log(2)
Dividing both sides of the equation by -6log(2), we get:
k = -1 / (-6log(2))
Now that we have the value of k, we can find the remaining mass after 2 years.
Plugging the values into the equation, we have:
D = D_0 * 2^(-kt)
D = D_0 * 2^(-k*2)
D = D_0 * 2^(-2k)
Since we want to find the percentage remaining, we can divide D by D_0 and multiply by 100:
Percentage remaining = (D / D_0) * 100
Plugging in the value of D and D_0, we get:
Percentage remaining = (D_0 * 2^(-2k) / D_0) * 100
Percentage remaining = 2^(-2k) * 100
Now, substituting the value of k we calculated earlier, we can find the percentage remaining after 2 years:
Percentage remaining = 2^(-2 * (-1 / (-6log(2)))) * 100
Simplifying further:
Percentage remaining = 2^(-2 / (-6log(2))) * 100
Percentage remaining ≈ 81.59%
Therefore, after 2 years, approximately 81.59% of the uranium remains.
To find the percentage remaining after 2 years, we need to use the decay equation D = D_0 * 2^(-kt), where:
- D is the current mass of uranium
- D_0 is the initial mass of uranium
- k is the decay constant
- t is the time duration in years
We are given that it takes 6 years for the mass of uranium to halve, which means that after 6 years, D = D_0 / 2.
To find the decay constant k, we can rearrange the equation as follows:
D = D_0 * 2^(-kt)
D_0 / 2 = D_0 * 2^(-6k)
Canceling D_0 from both sides:
1 / 2 = 2^(-6k)
Taking the logarithm of both sides to solve for -6k:
log(1/2) = log(2^(-6k))
log(1/2) = -6k * log(2)
Simplifying:
k = log(1/2) / (-6 * log(2))
Now that we have the value of k, we can find the remaining percentage after 2 years. Plugging in the values into the decay equation:
D = D_0 * 2^(-kt)
D = D_0 * 2^(-k * 2)
To calculate the percentage remaining, we divide D by D_0 and multiply by 100:
Percentage remaining = (D / D_0) * 100
Plugging in the values and calculating:
Percentage remaining = (D_0 * 2^(-k * 2) / D_0) * 100
Percentage remaining = 2^(-k * 2) * 100
Using the value of k we calculated earlier:
Percentage remaining = 2^(-k * 2) * 100
Therefore, to find the percentage remaining after 2 years, you can calculate 2^(-k * 2) and multiply by 100.