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.
(1 point)
Responses
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 radioactive material is:
y = 49 * (1/2)^(x/42)
To find how much radioactive material remains after 8 days, we substitute x = 8 into the equation:
y = 49 * (1/2)^(8/42)
Simplifying this expression:
y = 49 * (1/2)^(2/21)
y = 49 * (0.985171)
y ≈ 48.233 kg
Rounded to the nearest thousandth, the amount of radioactive material that remains after 8 days is approximately 48.233 kg.