$(20$ points$)$ Replacing a super$-$lightweight alloy seatpost on a

bicycle with a polymer$-$composite seatpost can save up to $\frac{1}{4}$

pound of weight for a bicycle $($but costs about $3$ x more$).$ I

broke several polymer$-$composite seatposts on my mountain

bike $($shown in photos below$)$ and did some analysis to better

understand the way the stresses are applied.

If we use a continuous and aligned fiber$-$reinforced polymer

composite, we can meet the minimum strength requirements if

the elastic modulus along the fiber direction is $60$ GPa and

perpendicular to the fiber direction is $8\mathrm{.}5$ GPa $.$ One seatpost

manufacturer $($AX Lightness Europa$)$ uses Kevlar KM$2$ aramid

fibers in a West Systems $105/209$ epoxy matrix. The properties

of the materials in the AX Lightness Europa seatpost and other

available materials are in the table below. Some important

questions arise:

a$.$ If I purchase this AX Lightness Europa seatpost, am I likely to

break it$?($That is$,$ will the elastic modulus criteria described

above be met? Justify your response with a calculation or series

of calculations in a spreadsheet$).$

b$.$ If the AX Lightness Europa composite seatpost is expected to

break, please select the combination of available materials from

the table below meeting the modulus requirements and

minimizing the weight. Again, use calculations and$/$or a

spreadsheet to justify your selection and include these in your

response.

$*$a medicinal plant native to the Amazon