The decay rate of the radioactive material is 0.00244 or 0.244%. To graph the function A(t) = 500e^(-0.00244t), plot it on a graphing utility or software. After 30 years, approximately 384.75 grams of radioactive material will be left. When 300 grams of the material are left, it will be approximately 34.38 years. The half-life of the material is approximately 284.64 years.
(a) The decay rate of the radioactive material can be determined from the given function A(t) = A0e^(kt), where A0 is the initial amount present and k is the decay constant or rate. In this case, we have A(t) = A0e^(-0.00244t). The decay rate is equal to the absolute value of the decay constant, which is 0.00244. So, the decay rate of the radioactive material is 0.00244 or 0.244%.
(b) To graph the function, you can use a graphing utility or software. Plot the function A(t) = 500e^(-0.00244t) on the vertical axis against time t (in years) on the horizontal axis.
(c) To find how much radioactive material is left after 30 years, substitute t = 30 into the given function A(t) = 500e^(-0.00244t). A(30) = 500e^(-0.00244*30) = 500e^(-0.0732) ≈ 384.75 grams.
(d) To find when 300 grams of the radioactive material will be left, set A(t) = 300 in the given function A(t) = 500e^(-0.00244t) and solve for t. 300 = 500e^(-0.00244t) ⇒ e^(-0.00244t) = 0.6 ⇒ -0.00244t = ln(0.6) ⇒ t ≈ 34.38 years.
(e) The half-life of the radioactive material can be determined by finding the value of t for which A(t) = A0/2. Set A(t) = 500e^(-0.00244t) = 250 and solve for t: 250 = 500e^(-0.00244t) ⇒ e^(-0.00244t) = 0.5 ⇒ -0.00244t = ln(0.5) ⇒ t ≈ 284.64 years. Therefore, the half-life of the radioactive material is approximately 284.64 years.