Differential equations (40 pts) Consider a certain radioactive isotope sample with an initial mass, M0. It is observed that the mass of radioactive sample decays with time. The rate of mass decay at time t is proportional to the mass M(t) at that moment. The constant of proportionality is the decay rate constant, r > 0. The rate of mass decay is given by dM(t) dt = −rM(t). (1) (a) (20 pts) Solve the differential equation (1) to determine the mass of the sample at a given time.

An engine, weighing 3000 n, is supported on a pedestal mount. it has been observed that the engine induces vibration into the surrounding area through its pedestal at the maximum operating speed. determine the stiffness of the dynamic vibration absorber spring in (n/m) that will reduce the vibration when mounted on the pedestal. the magnitude of the exciting force is 250 n, and the amplitude of motion of the auxiliary mass is to be limited to 2 mm note: in this question type-in right numbers, no decimals, no fractions, no unit. approximate to right number if needed