Question

given that vanadium-48 has a radioactive half-life of approximately 16 days, determine a so that \\(a(t) = a_0 a^t\\) describes the amount of vanadium-48 left after \\(t\\) days (as usual, \\(a_0\\) is the amount at time \\(t = 0\\)). round your answer to three decimal places.

Explanation:

Set up the half-life equation

$$ A(t) = A_0 a^t $$

$$ A(16) = \frac{1}{2} A_0 $$

Solve for the base \(a\)

$$ A_0 a^{16} = \frac{1}{2} A_0 $$

$$ a^{16} = 0.5 $$

$$ a = (0.5)^{\frac{1}{16}} $$

Calculate the numerical value

$$ a \approx 0.957603 $$

$$ a \approx 0.958 $$

Answer:

Given that vanadium-48 has a radioactive half-life of approximately 16 days, determine a so that \(A(t) = A_0 a^t\) describes the amount of vanadium-48 left after t days (as usual, \(A_0\) is the amount at time \(t = 0\)). Round your answer to three decimal places.

\(a \approx\) <blank>0.958</blank>