$($For Problems $1-2)$

Determine the properties of the combined sections shown below. Pls provide detailed and complete solution and final answer with numerical values

a$.$ Area

b$.$ Centroid

c$.$ Moment of inertia $($Ix and Iy$)$

$"$C$"=9$; $"$B$"=2$; $"$A$"=1$

d$.$ Section Modulus $($Sx and Sy$)$

e$.$ Radius of gyration $($rx and ry$)$

USE THE HIGHLIGHTED VALUES

f$.$ Section modulus at an axis of max moment of inertia $(S_{\text{m}\text{a}\text{x}\text{i}})$

g$.$ Radius of gyration at an axis of min moment of inertia $(r_{\text{m}\text{i}\text{n}\text{I}})$

$\backslash$table$[["$B$",$Channel Section,"C$",$Wide Flange Section,"A$",$Angle Section$],[0,$C$15$X$50,0,$W$30$x$99,0,$L$65\backslash$times $50$x$8],[1,$C$15\backslash$times $40,1,$W$30$x$90,1.,$L$65\backslash$times $50$x$10],[2,-$ C$12\backslash$times $30,2,$W$27$x$161,2,$L$75\backslash$times $50$x$6],[3,$C$12$x$25,3,$W$27\backslash$times $146,3,$L$75\backslash$times $50$x$8],[4,$C$10$x$25,4,$W$24$x$176,4,$L$90$x$75$x$6],[5,$C$10\backslash$times $20,5,$W$24$x$162,5,$L$90$x$75\backslash$times $8],[6,$C$9$X$15,6,$W$21\backslash$times $166,6,$L$125\backslash$times $75\backslash$times $8],[7,$C$9$X$13\mathrm{.}4,7,$W$21\backslash$times $147,7,$L$125\backslash$times $75\backslash$times $10],[8,$C$7$x$14\mathrm{.}75,8,$W$18$x$211,8,$L$150$x$90\backslash$times $10],[9,$C$7$x$12\mathrm{.}25,9,-$ W$18$x$192,9,$L$150$x$90\backslash$times $12]]$

Use the Steel Manual for the properties of the above sections.

Channel section and angular section with longer leg vertical

Two identical Channel Sections placed back to back with a clear distance of $25$ mm $.$

Given a Wide Flange section, $($see above table for your WF section$)$ Determine the following:

a$.$ Maximum moment of inertia in $m{m}^4.$

b$.$ Minimum moment of inertia in $m{m}^4.$

c$.$ Moment of inertia in $m{m}^4$ with respect to the $x^{\text{'}}-$axis, rotated counterclockwise $55$ from the $x-$axis.

d$.$ Product moment of inertia in $m{m}^4$ with respect to the $x^{\text{'}}-$axis.

e$.$ Maximum product moment of inertia in $m{m}^4.$

f$.$ Minimum product moment of inertia in $m{m}^4.$