$1$ Thin$-$walled Approximation

Consider a hollow beam with square cross section. Each side of the cross section has outer length a and

wall thickness $t$; i$.$e$.,$ the inner opening is $(a-2t)$ by $(a-2t).$ Using the thin$-$walled assumption $(ta),$

we get $I_{}=I_{yy}$~~$2a^3\frac{t}{3}.$ Compute the exact moment of inertia $I_{}=I_{yy}$ of the cross section; i$.$e$.,$ do not

simply assume that $ta.$ Then plot the ratio of the exact moment of inertia to the approximate moment

of inertia for a range of $\frac{a}{t}in[2,200].$ What percent error does the approximation introduce for $\frac{a}{t}=10,$

$25,$ and $100?$