A baseball diamond is a square with an area of 8100 square feet. The length of the diagonal of any square is equal to \(\sqrt2\) times its side length. Find the distance from home plate to second base (the length of the diagonal) to the nearest hundredth of a foot.

\( a-the\ sides\ of\ the\ square\\\\A_{\fbox{}}=a^2;\ A_{\fbox{}}=8100ft^2\\\\a^2=8100ft^2\\\\a=\sqrt{8100ft^2}\\\\a=90ft\\-\\The\ diagonal\ of\ the\ square: d=a\sqrt2\\-\\The\ distance\ from\ home\ plate\ to\ second\ base:\\\\d=90\sqrt2\ ft\approx(90\times1.414)ft=\boxed{127.26ft} \)

\( A_{square}=(the\ side\ of\ the\ square)^2\\\\A_{square}=8100\ ft^2\ \ \ \Leftrightarrow\ \ \ A_{square}=(90\ ft)^2\\\\(the\ side\ of\ the\ square)^2=(90\ ft)^2\ \ and\ \ \ the\ side\ of\ the\ square>0\\\\the\ side\ of\ the\ square=91\ ft\\\\the\ length\ of\ the\ diagonal= \sqrt{2} \cdot the\ side\ length\ (of\ the\ square)\\\\the\ length\ of\ the\ diagonal= \sqrt{2} \cdot 90\ ft\\\\ \sqrt{2} =1.4142135623.\approx1.4142\\\\ \)

\( \Rightarrow\ \ \ the\ length\ of\ the\ diagonal\approx1.4142\cdot90\ ft=127.278\ ft \)


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