Carbon dating an important tool in archeological research is radiocarbon dating, developed by the american chemist willard f. libby. this is a means of determining the age of certain wood and plant remains, hence of organic materials such as animal or human bones or artifacts found buried at the same levels. radiocarbon dating is based on the fact that some wood or plant remains contain residual amounts of carbon-14 (c-14)-a radioactive isotope of carbon. this isotope is accumulated during the lifetime of the plant and begins to decay at its death. since the half-life (i. e., the amount of time it takes for a quantity of radioactive material to decay to one-half of its original amount) of c-14 is very long (approximately 5730 measurable amounts of c-14 remain after many thousands of years. if even a tiny fraction of the original amount of c-14 is still present, then by appropriate laboratory measurements the proportion of the original amount of c-14 that remains can be accurately determined. in other words, if q(t) is the amount of c-14 at time t and qo is the original amount, then the ratio q(t)/qo can be determined, at least if this quantity is not too small. (fyi: the present techniques permit the use of this method for time periods of 50,000 years or more! ) 1 (a) (2 points) assuming that q satisfies the differential equation dq =-rq dt find an expression for q(t) at any time t (in years), if q(0) = qo. that is, solve the corresponding ivp. (b) (1 point) determine the decay constant r for c-14 to three significant digits. also, what is the unit of this decay constant? (c) (2 points) suppose that certain remains are discovered in which the current residual amount of c-14 is 15% of the original amount. determine the age of these remains to the nearest year.