The half-life of a certain radioactive material is 78 hours. An initial amount of the material has a mass of 790 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 18 hours. Round your answer to the nearest thousandth.
(1 point)
The equation for exponential decay is given by:
A(t) = A₀ * e^(-kt)
Where:
A(t) = amount of material at time t
A₀ = initial amount of material
k = decay constant
t = time
We are given that the half-life of the material is 78 hours. The decay constant can be found using the formula:
k = ln(2) / half-life
k = ln(2) / 78
We are also given that the initial amount of material is 790 kg. So, the exponential decay function is:
A(t) = 790 * e^(-kt)
To find the amount of material remaining after 18 hours, substitute t = 18 into the equation:
A(18) = 790 * e^(-k*18)
Rounded to the nearest thousandth:
A(18) ≈ 790 * e^(-k*18) ≈ 790 * e^(-ln(2)*18/78) ≈ 790 * e^(-0.00873) ≈ 787.030 kg
Therefore, after 18 hours, approximately 787.030 kg of radioactive material remains.