years. When air is released from an inflated balloon it is found that the rate of decrease of the volume of the balloon is proportional to the volume of the balloon. This can be represented by the -kv, where v is the volume, t is the time and k is the constant of dy differential equation dr proportionality. (a) If the initial volume of the balloon is vo, find an expression, in terms of k, for the volume of the balloon at time 1. Vo (b) Find an expression, in terms of k, for the time when the volume is

Newton’s law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). We let (t) denote time, T(t) be the temperature of the object at time (t), (T0 = T(0)) (the object’s temperature at (t = 0)), and Tomg be the constant ambient temperature. Set u(t) = T(t) - Tomg and show from Newton’s law of cooling that (u) satisfies the initial value problem (*) u’(t) = ku(t), u(0) = T0 - Tomg       for a constant (k). Solve the initial value problem (*), i.e., find u(t)) (expressed in terms of (k), T0, and Tomg . Find T(t).

The half-life of a substance is found by using the equation A(t) = Aoe-kt where A(t) is the amount present at time t (years), k is the decay rate, and Ao is the initial amount of the substance present. Find the half-life of the substance. (Round your answer to the nearest year.) A radioactive substance whose decay rate is 0.6% per year yr