A contractor is planning to use single$-$engine scrapers to move $90,000$ bank cubic yards of soil weighing $2,850$ pounds per bank cubic yard. The percent swell for the soil has been determined to be $23\%.$ To load the scrapers their rated heaped capacity, the contractor plans to use tracked dozers. The $3,000-$foot haul road from the borrow site to the fill area is fairly well maintained with a rolling resistance estimated to be $90$ pounds per ton. The coefficient of traction is estimated to be $0\mathrm{.}5.$ The haul road slopes down at $1\%$ slope when traveling from the borrow site to the fill area. Ownership and operating cost for each scraper is about $$200$ per hour and about $$135$ per hour for the pusher dozer. A scraper operator earns $$60$ per hour, and a dozer operator earns $$58$ per hour. Job efficiency is estimated to be $45$ minutes per hour, and the job conditions are considered average. How many scrapers should the contractor plan to use one this job, if he wants to minimize idle time for the pusher dozer?