You are tasked with determining the most cost$-$effective haul fleet for loading trucks using a hoe equipped with a $3-$cubic$-$yard bucket. The working efficiency of the hoe is $50$ minutes per hour, and the material loaded is dry gravel. The digging face has an average height of $60\%$ of the hoe's maximum digging depth, and the angle of swing is always less than $60.$ You have two truck size options: $1.$ A $20-$cubic$-$yard truck, with an operating cost of $$55/$hr $2.$ A $24-$cubic$-$yard truck, with an operating cost of $$75/$hr The operator of the hoe costs $$68/$hr$,$ and the round$-$trip time of either option $($above$),$ exclusive of load time, is $5$ minutes. Which truck size $(20$ lcy or $24$ lcy$)$ results in the lowest cost when considering the production rate?