= Solution
On a logarithmic strength–density <materials selection chart>, constant $M_A$ gives $\sigma_f=M_A\rho$, a guideline of slope one; constant $M_B$ gives $\sigma_f=M_B^2\rho^2$, of slope two. Moving each guideline upwards selects better candidates. \b[A suitably qualified longitudinal carbon-fibre-reinforced polymer is a promising candidate for both designs.] Composite families occupy favourable high-strength/low-density regions in published course selection charts, but a beam must meet its compressive as well as tensile allowable.<Examples of Materials Selection by Wright State University>[Unit 5, strength–density charts and composite selection examples; family comparisons are screening evidence, not universal longitudinal compression data]{ref}
For scale, an explicitly illustrative candidate with $\sigma_{\rm lim}=1\,\mathrm{GPa}$ in both required loading senses and $\rho=1600\,\mathrm{kg\,m^{-3}}$ would have $M_A=6.25\times10^5\,\mathrm{m^2s^{-2}}$ and $M_B=19.76\,\mathrm{Pa^{1/2}m^3kg^{-1}}$, exceeding the best supplied materials. These are assumed screening values, not measured properties of a specified product.
\Image[/past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2022/paper-1-selection.png]
{title=Strength–density selection with the two cantilever merit-index guidelines}
{description=Original chart of the three supplied candidates and one explicitly assumed composite design point. Pine uses its compression limit; the figure is not a complete material-family database.}
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