The paralle$-$axis theorem can be used to find an area's moment of inertia about any axis that is parallel to an axis that passes through the centroid and whose moment of inertia is known. It $x^{\text{'}}$ and $y^{\text{'}}$ are the axes

that pass through an area's centroid, the paralle$-$axis the about the $y$ axis, and the polar moment of inertia is expressed tor the moment about the $x$ axis, moment

$I_z\frac{?}{b}=ar(I{)}_z+A{d}_v^2$

$I_y\frac{?}{b}=ar(I{)}_v+A{d}_z^2$

$J_O\frac{?}{b}=ar(J{)}_{C^{\text{'}}}+A{d}^2$

where $I_z$ is the area's moment of ineria about the noncentroidal $x$ axis, $\frac{?}{\text{b}\text{a}\text{r}}(I{)}_x$ is the moment of inertia about the centroidal $x$ axis, $A$ is the total area, $d_y$ is the perpendicular distance in the $y$ direction between the centroid and the $x$ axis, $I_y$ is the area's moment of ineria about the noncentroidal $y$ axis, $\frac{?}{\text{b}\text{a}\text{r}}(I{)}_y$ is the moment of inertia about the centroidal $y$ axis, $d_2$ is the perpendicular distance in the $x$ direction between the centroid and the $y$ axis, $J_O$ is the polar moment of inertia about some noncentroidal point, $\frac{?}{\text{b}\text{a}\text{r}}(J{)}_C$ is the polar moment of inertia about the centroid, and $d$ is the distance between the points $O$ and $C.$

As shown, an area has a centroid located at point $C.($Elgure $1)$ The area's moment of inertia about the $x^{\text{'}}$ and $y^{\text{'}}$ centroidal axes are tilde$(I{)}_{z^{\text{'}}}$ and $\frac{?}{\text{b}\text{a}\text{r}}(I{)}_{y^{\text{'}}},$ respectively, Consider the point $O_1,$ which is located, relative to point $G$ a distance dat in the nepative $x$ direction and $d_{y1}$ in the positive $y$ direction. Given that $I_{x1}$ is the moment of the area about the $x_1$ axis, $I_{y1}$ is the moment of the area about the $y_1$ axis, $\frac{?}{\text{b}\text{a}\text{r}}(J{)}_C$ is the polar moment of inertia about the centroid, and $J_{01}$ is the polar moment of inertia aison point $O_1,$ sort the following expressions into the three categories below.

Drag the appropriate items to their respective bins.

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Part B

As shown, a rectangle has a base of $b=8\mathrm{.}40ft$ and a height of $h=1\mathrm{.}20ft.($Figure $2)$ The rectangle's bottom is located at a distance $y_1=1\mathrm{.}30ft$ from the $x$ axis, and the rectangle's left edge is located at a distance $x_1=2\mathrm{.}90ft$ trom the $y$ axis. What are $I_x$ and $I_y,$ the area's moments of inertia, about the $x$ and $y$ axes, respectively?

Express your answers numerically in biquadratic feet $($feet to the fourth power$)$ to three significant figures separated by a comma.

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$I_x,I_y=37\mathrm{.}6,567f{t}^4$

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Part C

The semicircle shown $($Eigure $3)$ has a moment of inertia about the $x$ axis of $77\mathrm{.}0f{t}^4$ and a moment of inertia about the $y$ axis of $77\mathrm{.}0f{t}^4.$ What is the polar moment of inertia about point $C($the centroid$)?$ Express your answer numerically in biquadratic feet $($feet to the fourth power$)$ to three significant figures.

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$$

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$154\mathrm{.}0$

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Answer part C