Final answer:
To address the questions about radioactive decay, the original amount present was 0.0125 milligrams, while the times when half of the isotope has transformed and when only 0.005 milligrams remain can be determined by solving exponential equations based on the given decay function.
Step-by-step explanation:
The question relates to the concept of radioactive decay and how a radioactive isotope transforms into a more stable isotope over time, described mathematically by a decay function. The equation given is A(t) = 0.0125e^{-t/500}, where A represents the amount of the isotope in milligrams and t is the time in seconds.
a) The original amount of the isotope can be found by determining the value of A when t is 0. Plugging in 0 for t in the equation A(0) = 0.0125e^{0} yields A(0) = 0.0125. Therefore, originally there were 0.0125 milligrams of the isotope present.
b) To find when half of the original amount will be transformed, we need to set A(t) to half of the original amount, which is 0.00625 milligrams, and solve for t. This gives us the equation 0.00625 = 0.0125e^{-t/500}. Solving for t will provide the time at which half of the isotope has decayed.
c) To find out when there will be 0.005 milligrams of the isotope remaining, we set A(t) to 0.005 and solve for t. The resulting equation is 0.005 = 0.0125e^{-t/500}, which, when solved, will give us the time it takes to reach 0.005 milligrams.