91Ó°ÊÓ

Determine the time of death if the temperature of a corpse is \(79^{\circ} \mathrm{F}\) when discovered at \(3: 00\) p.m. and \(68^{\circ} \mathrm{F} 3\) h later. Assume that the temperature of the surroundings is \(60^{\circ} \mathrm{F}\) (normal body temperature is \(98.6^{\circ} \mathrm{F}\) ).

Suppose that mold grows at a rate proportional to the amount present. If there are initially \(500 \mathrm{~g}\) of mold and \(6 \mathrm{~h}\) later there are \(600 \mathrm{~g}\), determine the amount of mold present after one day. When is the amount of mold \(1000 \mathrm{~g}\) ?

Solve the Logistic equation, \(d y / d t=\alpha y(1-\) \((1 / K) y\) ), by viewing it as a Bernoulli equation and then solve the resulting linear equation by using an integrating factor rather than the method of undetermined coefficients that is illustrated in the examples.

(Escape Velocity) Suppose that a rocket is launched from the Earth's surface. At a great (radial) distance \(r\) from the center of the Earth, the rocket's acceleration is not the constant \(g\). Instead, according to Newton's law of gravitation, \(a=k / r^{2}\), where \(k\) is the constant of proportionality ( \(k>0\) if the rocket is falling toward the Earth; \(k<0\) if the rocket is moving away from the Earth). (a) If \(a=-g\) at the Earth's surface (when \(r=R\) ), find \(k\) and show that the rocket's velocity is found by solving the initial value problem \(d v / d t=-g R^{2} / r^{2}, v(0)=v_{0}\). (b) Show that \(d v / d t=v d v / d r\) so that the solution to the initial value problem \(v d v / d r=\) \(-g R^{2} / r^{2}, v(R)=v_{0}\) is \(v^{2}=2 g R^{2} / r+v_{0}^{2}-\) \(2 g R\). (c) If \(v>0\) (so that the rocket does not fall to the ground), show that the minimum value of \(v_{0}\) for which this is true (even for very large values of \(r\) ) is \(v_{0}=\sqrt{2 g R}\). This value is called the escape velocity and signifies the minimum velocity required so that the rocket does not return to the Earth.

Find the equilibrium solution to \(d v / d t=\) \(-g-(c / m) v^{2}\). What is the limiting velocity?

Consider a solution to the logistic equation with initial population \(y_{0}\) where \(0