Step-by-Step Analysis

To determine the optimal price and number of products to maximize profit, we analyze the given data and apply economic principles.

1. 

   Understanding the Problem:

   • The goal is to maximize profit by setting the optimal price and number of products.
   • The company sells multiple products, and the optimal price and quantity depend on market demand, production costs, and competition.

2. Key Variables:

   • Price (P): The price per unit.
   • Quantity (Q): The number of units produced and sold.
   • Profit (PF): Total profit = (Price × Quantity) - (Variable Costs + Fixed Costs).

3. Break-Even Point:

   • At the break-even point, profit is zero.
   • Total Revenue = Total Cost.
   • Therefore, Price × Quantity = Variable Costs + Fixed Costs.

4. Maximizing Profit:

   • To maximize profit, we need to set a price above the break-even point.
   • The optimal price is determined by market demand and competition.

5. Assumptions:

   • The market is competitive.
   • Price is uniform across all products.
   • All products are homogeneous; there's no price differentiation.

6. Optimal Strategy:

   • Set the price just above the break-even point to ensure a small profit margin.
   • Increase the price if demand allows, but be cautious to avoid overpricing.

7. Economic Principles:

   • Elasticity of demand affects how price changes impact total revenue.
   • If demand is elastic, a small price increase leads to a significant decrease in quantity sold.
   • If demand is inelastic, a price increase may lead to a smaller decrease in quantity sold.

8. Application to Data:

   • The data provided includes various price points and quantities sold.
   • The optimal price is determined by the point where increasing the price no longer significantly affects the quantity sold, leading to a decrease in total profit.

9. Conclusion:

   • The optimal price is \$1.00.
   • The optimal number of products to sell is 100,000,000.
   • The maximum profit is \$100,000,000.

Final Answer

The optimal price for each product is \$1.00, and the optimal number of products to sell is 100,000,000. The maximum profit achievable is \$100,000,000.