The figure below shows a TEM micrograph of a rubber toughened polystyrene, PS$,$ in which the rubber particles are indicated by arrows. The rubber, which has a Tg of $-60$oC$,$ can decrease the Tg of the PS $($which has a Tg of $100$oC$),$ however.

a$)$ The fracture toughness of the HIPS sample is $15$ kJ$/$m$2$ vs$.0\mathrm{.}4$ for pure PS$.$ A typical method for measuring this property involves placing a small crack in the sample, applying a stress and measuring its resistance to brittle fracture, as seen in the figure below. Discuss the mechanism$($s$)$ by which the rubber increases the fracture toughness of HIPS relative to pure PS$.$

b$)$ How much rubber must be added in order to decrease the Tg of PS

to $65$oC$?$