An electronic device that dissipates(generates) \(30 \mathrm{~W}\) is used infrequently. Its maximum allowable operating temperature is limited to \(75^{\circ} \mathrm{C}\); as soon as \(75^{\circ} \mathrm{C}\) is reached, the device must be shut off. The device and an attached heat sink have a combined mass of \(0.25 \mathrm{~kg}\), surface area of \(56.3 \mathrm{~cm}^{2}\), volume of \(50 \mathrm{~cm}^{3}\), a thermal conductivity of \(2 \mathrm{~W} / \mathrm{m} \mathrm{K}\), and an effective specific heat of \(800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). If the device is initially at a uniform temperature of \(25^{\circ} \mathrm{C}\) in air at \(25^{\circ} \mathrm{C}\) with a heat-transfer coefficient of \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) :

a. Derive the differential equation governing the temperature of the device assuming we can use a lumped capacitance formulation.

b. Determine the steady-state operating temperature (in \({ }^{\circ} \mathrm{C}\) ). (You can solve the equation in several ways. The easiest is to define a new temperature variable \(\theta=T-\Sigma\) Constants.)

c. Determine the time required to reach the maximum operating temperature (in s).

d. If a heat sink is to be added to the device so that the operating time is to be doubled, what is the total area needed? Assume the mass-to-area ratio of the added material is the same as that of the original device.