Thread: optimization - fixed and variable cost?

Murray can hire a dump truck for $40/h (deliver included). The truck takes 30 min to deliver a load of topsoil and return. It would take 40h for one person to load the truck with topsoil. Laborers get $10/h whether they are loading the truck or standing idle while they await its return. How many laborers should be hired to minimize the cost per load. What is the minimum cost?

Variable costs: If x is the number of workers, it will take 40/x hours and cost 1600/x to hire the truck while its being filled. It will cost 5x to have x workers stand idle for 30 minutes at $10/hr. Cost per load will be minimized when variable costs are minimized. So you need to minimize 1600/x + 5x. Take the derivative, set it equal to zero, solve for x.
Fixed costs: $400 in labor to fill truck; $20 of truck hire during return trip.