A shaft is ${3\mathrm{.}00}^{\text{'}\text{'}}$ in nominal diameter. A gear is going to be mounted on the shaft and needs a key to transmit torque to the shaft. The gear will transmit $500ft-lb$ of torque. The safety factor for this application must be $1\mathrm{.}5$ minimum. If the key has a yield strength of $120$ kpsi $,$ size the key and analyze the stress on the key to make sure it is of sufficient size. $(10$ points$)$