A watermelon is taken out of the refrigerator at a uniform temperature of \(3^{\circ} \mathrm{C}\) and is exposed to \(32^{\circ} \mathrm{C}\) air. Assume that the watermelon can be approximated as a sphere and the temperature of the watermelon is uniform. The estimated parameters are density \(ho=120 \mathrm{~kg} / \mathrm{m}^{3}\), diameter \(D=35 \mathrm{~cm}\), specific heat capacity \(c=4200 \mathrm{~J} /\left(\mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right)\), and heat transfer coefficient \(h=15 \mathrm{~W} /\left(\mathrm{m}^{\left.2 .{ }^{\circ} \mathrm{C}\right) \text {. }}\right.\)

a. Derive the differential equation relating the watermelon's temperature \(T(t)\) and the air temperature.

b. Using the differential equation obtained in Part (a), construct a Simulink block diagram and find the temperature of the watermelon.

c. Build a Simscape model of the system.

d. Based on the simulation results obtained in Parts

(b) and (c), how long will it take before the watermelon is warmed up to \(20^{\circ} \mathrm{C}\) ?