The half-life of a certain radioactive material is 42 days. An initial amount of the material has a mass of 49 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 8 days. Round your answer to the nearest thousandth.
y = 49one-half superscript 42 x baseline; 0 kg
y = 49 Image with alt text: one-half superscript 42 x baseline ; 0 kg
y = one-halfleft-parenthesis start fraction 1 over 49 end fraction right-parenthesis superscript start fraction 1 over 42 end fraction x baseline; 0.238 kg
y = Image with alt text: one-half Image with alt text: left-parenthesis start fraction 1 over 49 end fraction right-parenthesis superscript start fraction 1 over 42 end fraction x baseline ; 0.238 kg
y = 49one-half superscript start fraction 1 over 42 end fraction x baseline; 42.940 kg
y = 49 Image with alt text: one-half superscript start fraction 1 over 42 end fraction x baseline ; 42.940 kg
y = 2left-parenthesis start fraction 1 over 49 end fraction right-parenthesis superscript start fraction 1 over 42 end fraction x baseline; 0.953 kg
The correct exponential function that models the decay of the material is y = 49 * (1/2)^(x/42), where x represents the number of days.
To find how much radioactive material remains after 8 days, substitute x = 8 into the equation:
y = 49 * (1/2)^(8/42)
y ≈ 49 * (1/2)^0.19047619047619047
y ≈ 49 * 0.9070286425897852
y ≈ 44.469 kg
Therefore, after 8 days, approximately 44.469 kg of radioactive material remains. Rounded to the nearest thousandth, the answer is 44.469 kg.