8. [25 pts.] Suppose that a hot metal bar with absolute temperature T(t) at time t...

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8. [25 pts.] Suppose that a hot metal bar with absolute temperature T(t) at time t cools by emitting thermal radiation. According to the Stefan- Boltzmann law, the bar radiates heat energy at rate proportional to T¹. If the initial temperature of the bar is To > 0, then T(t) satisfies the initial value problem dT -kT4, dt T(0) = To, where k 0 is a constant. (a) Solve for T(t). (b) Let t₁/2 be the time it takes for the bar to cool to half its initial temper- ature. Find an expression for t₁/2 in terms of k and To. (c) Does t₁/2 increase, decrease, or stay the same as To increases? How does this compare with the behavior of t1/2 for an object that cools according to Newton's law of cooling (dT/dt = -kT)? 8. [25 pts.] Suppose that a hot metal bar with absolute temperature T(t) at time t cools by emitting thermal radiation. According to the Stefan- Boltzmann law, the bar radiates heat energy at rate proportional to T¹. If the initial temperature of the bar is To > 0, then T(t) satisfies the initial value problem dT -kT4, dt T(0) = To, where k 0 is a constant. (a) Solve for T(t). (b) Let t₁/2 be the time it takes for the bar to cool to half its initial temper- ature. Find an expression for t₁/2 in terms of k and To. (c) Does t₁/2 increase, decrease, or stay the same as To increases? How does this compare with the behavior of t1/2 for an object that cools according to Newton's law of cooling (dT/dt = -kT)?

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