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$($a$)$ Some authors have argued that the plates on a Stegosaurus are used as a radiative device to cool the animal $($or as a passive heat sink to warm it$).$ Assume that this is true. Individuals of Stegosaurus could grow to be $9$ m in length. Assume that the closely related Kentrosaurus, which is ornamented with large spikes instead of plates, is $\backslash (80\backslash \%\backslash )$ the linear dimensions of Stegosaurus $($the bodies only, not any of the ornamentation$).$ As a simplification, model the entire body of these animals as a sphere $-$ ignore the head and tail. What is the relative heat flux of Kentrosaurus, as a percentage of that of Stegosaurus?

$($b$)$ Also assume that the spikes on Kentrosaurus and on the tail of Stegosaurus are non$-$vascular and therefore do not affect temperature regulation. Model the plates of Stegosaurus as a set of $6$ triangles, all of equal size and with a base equal to their height. Also, assume that they are acting as thin sheets $-$ adding surface area, but negligible volume to the system. How tall do these plates need to be in order for Stegosaurus to have the same heat flux as Kentrosaurus?

Recall that the circumference of a circle is $\backslash (2\backslash$pi r $\backslash ),$ the area of a circle is $\backslash (\backslash$boldsymbol$\{\backslash$pi$\}\backslash$mathbf$\{$r$\}^\{\backslash$mathbf$\{2\}\}\backslash ),$ the surface area of a sphere is $\backslash (4\backslash$pi r$^\{2\}\backslash ),$ and the volume of a sphere is $(4/3)\backslash (\backslash$pi r$^\{3\}\backslash ).$