# 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%

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## 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.

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## 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.