V$13-12.$ A thermal pebble$-$bed reactor has a $5-$ft$-$diameter spherical core. It uses $1-$in$.-$diameter spherical fuel pebbles made of $3$ percent enriched $\backslash (\backslash$mathrm$\{$UO$\}\_\{2\}\backslash ).$ The pebbles are randomly packed in the core so that they occupy about twothirds of the core volume. The maximum thermal$-$neutron flux is $\backslash (10^\{13\}\backslash ).$ Neglect the effects of extrapolated length and cladding, and determine $($a$)$ the maximum heat produced by a single fuel element and $($b$)$ the maximum fuel temperature in the same fuel element if its surface temperature is $\backslash (1000^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).($c$)$ Is this the maximum fuel temperature in the core? Why? Use $550$ b for fission cross section.