The half life of uranium is 69 years. How long will it take uranium to reach 25% of its
original value?
a) Find k, round to 2 decimals, show all work
b) Find solution, round to 2 decimals, show all work
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To solve this problem, we can use the formula for exponential decay:
N(t) = N0 * e^(-kt)
where:
N(t) = remaining value at time t
N0 = initial value
k = decay constant
t = time
We are given that the half-life of uranium is 69 years, which means that N(t) will be half of its initial value N0 after 69 years. In other words, N(t) / N0 = 0.5.
To find k, we can substitute these values into the equation:
0.5 = e^(-k * 69)
Taking the natural logarithm (ln) on both sides:
ln(0.5) = -k * 69
Now we can solve for k:
k = ln(0.5) / -69
k ≈ 0.01
a) The value of k is approximately 0.01.
Next, we need to find the time it takes for uranium to reach 25% of its original value, which means N(t) / N0 = 0.25.
Substituting into the equation:
0.25 = e^(-0.01 * t)
Taking the natural logarithm (ln) on both sides:
ln(0.25) = -0.01 * t
Now, solve for t:
t = ln(0.25) / -0.01
t ≈ 69.31 years
b) It will take approximately 69.31 years for uranium to reach 25% of its original value.
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