Two precious metal spheres initially at a temperature Ti = 800 K are placed (at different times) in a water bath held at T[infinity] = 300 K. A gold sphere of diameter Dg = .01 m experiences a convective heat coefficient hg = 1200 W m^2·K , while a silver sphere of diameter Ds = .06 m experiences hs = 3000 W m^2·K . Using the thermal properties evaluated at the initial temperature and the radius as the characteristic length:

a. Determine equations for the gold and silver spheres that describe their outside surface temperatures, Ts, G(t) and Ts, s(t), and center temperatures, To, c(t) and To, s(t), respectively.
b. Determine how long it takes the surface of each sphere to reach a temperature Ts, f = 500 K.
c. On the same set of axes, plot Temperature vs. Time for all the surface and center temperature trends in part (a). Include a title, legend, and axis labels with units. Set the limits of your time axis to 30 seconds, and temperature axis from T[infinity] to Ti.
d. Indicate on your plot where the surface temperature of the spheres cross 500 K verifying your answers to part (b).