Suppose you want to fence a rectangular area for your dog. You will use the house as one of the four sides. Since the house is 40 ft wide, the length $\ell$ needs to be no more than 40 ft. You plan to use at least 150 ft of fencing. Graph the following system to find possible dimensions for the rectangle.

$$\begin{array}{l}{\ell \leq 40} \\ {\ell+2 w \geq 150}\end{array}$$

A heat engine operating in outer space may be assumed equivalent to a Carnot engine operating between reservoirs at temperatures $T_H$ and $T_C.$ The only way heat can be discarded from the engine is by radiation, the rate of which is given (approximately) by: $\left|\dot{Q}_{C}\right|=k A T_{C}^{4}$ where k is a constant and A is the area of the radiator. Prove that, for fixed power output $|\dot{W}|$ and for fixed temperature $T_{H},$ the radiator area A is a minimum when the temperature ratio $T_{C} / T_{H}$ is 0.75

Define the coefficient of viscosity. Use kinetic theory to show that the coefficient of viscosity of a gas is given, with suitable approximations, by $\eta=K \rho\langle c\rangle \lambda$ where $\rho$ is the density of the gas, $\lambda$ is the mean free path of the gas molecules, $\langle c\rangle$ is their mean speed, and K is a number which depends on the approximations you make. In 1660 Boyle set up a pendulum inside a vessel which was attached to a pump which could remove air from the vessel. He was surprised to find that there was no observable change in the rate of damping of the swings of the pendulum when the pump was set going. Explain his observation in terms of the above formula. Make a rough order of magnitude estimate of the lower limit to the pressure which Boyle obtained; use reasonable assumptions concerning the apparatus which Boyle might have used. [The viscosity of air at atmospheric pressure and at 293 K is $18.2 \mu \mathrm{N} \mathrm{s} \mathrm{m}^{-2}$.] Explain why the damping is nearly independent of pressure despite the fact that fewer molecules collide with the pendulum as the pressure is reduced.