Consider a customer basis whose willingness$-$to$-$pay for a mechanic pencil is described by a continuous Unif$(0,20)$ distribution. The potential market for this product is $30,000$ units, which corresponds to the number of customers that would acquire the product in case its price was zero. Assume that product quantities can take continuous values.

a$)$ Formulate a linear demand function that captures the willingness$-$to$-$pay distribution of this market. How many customers would buy the product at p$=8?$ And at p$=20?$

b$)$ Provide a managerial problem definition for the optimization problem that maximizes the seller$$s revenues. Describe the objective function, decision variables and constraints. Assume that the unit cost is zero, and that the seller can hold infinite inventory. Formulate the problem in Excel. Plot revenues as a function of price, and use Excel Solver to get the optimal solution. Then, answer the questions below:$$i$)$ Is this a linear program?

ii$)$ Whatistheoptimalprice? iii$)$ What is the optimal revenue?