The total fixed costs per annum for a company that makes one product are $\$ 100,000$, and a variable cost of $\$ 64$ is incurred for each additional unit produced and sold over a very large range of outputs.

The current selling price for the product is $\$ 160$. At this price, 2,000 units are demanded per annum.

It is estimated that for each successive increase in price of $\$ 5$ annual demand will be reduced by 50 units. Alternatively, for each $\$ 5$ reduction in price, demand will increase by $\$ 50$ units.

Required:

(a) Calculate the optimum output and price, assuming that if prices are set within each $\$ 5$ range there will be a proportionate change in demand.

(b) Calculate the maximum profit.