A factory employs50workers whoproducehandcraftedtoys. Each worker can produce 5 toys in one hour. Working hours are flexible, so that each worker is free to choose how many hours he/she wants to work. For each worker, the effort cost of working is 5h², where h is the number of hours worked by this worker. The factory pays each worker a bonus b for each toy produced by that worker. Each worker's utility equals his/her own income minus the effort cost of working. Each worker chooses the number of hours that maximize his/her own utility.

The factory is able to sell each toy for a price P=20. In addition to the bonus paid to workers, the factory also incurs a cost for the material used to produce the toys. The cost of the material equals 0.002Q², where Q is to total quantity of toys produced by all workers in the factory.

Find the bonus b that maximizes the factory's profit.Hint: use q to represent the production of each worker and Q to represent the total production in the factory.

The factory will pay the workers a bonus b=?

for each toy produced.

Each worker will optimally choose to work h=?

hours.

The factory will produce a total of Q=?

toys.