Part A

An engineer is designing a structural beam that contains a lengthwise conduit that will be used for routing cable. As shown, the cross section of the beam has the dimensions $a=0\mathrm{.}740ft,b=1\mathrm{.}14ft,c=0\mathrm{.}210ft,$ and $d=0\mathrm{.}420ft.($Figure $1)$ The bearn's centroid is located at point $C(3\mathrm{.}33{10}^{-2}ft,2\mathrm{.}16{10}^{-2}ft).$ The engineer wishes to determine the beam's principal moments of inertia and the onentation of the principal axes using Mohr's circle. First, find the beam's moment of inertia and product of inertia What are the beam's moments of inertia, $I_x$ and $I_y,$ about the centroidal $x^{\text{'}}$ and $y$ axes, respectively? What is the beam's product of inertia about its centroid?

Express your answers numerically in feet to the fourth power to three significant figures separated by commas.

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