Problem $5(4$ points$)$: A spur gear pair is designed to have a reduction ratio of $0\mathrm{.}25,$ with a center distance of $7\mathrm{.}5$ inches. The pressure angle is $22\mathrm{.}5$ degrees. The pinion has $12$ teeth. Based on the proposed design,

a$)$ Find diametral pitch, pitch circle, addendum $($outside$)$ circle and dedendum $($root$)$ circle radii for the two gears.

b$)$ Determine contact ratio for the gear pair.

c$)$ For the same number of teeth on pinion and gear and same pressure angle, pitch radius of the pinion is reduced by $4\%$ from its value in part $($a$).$ For the new design, find pitch radii, center distance, diametral pitch, and new contact ratio.