The daily quantity demanded of milk in a regional market, measured in 1,000 's of gallons, can be represented during the summer months as the outcome of the following random variable:

\(Q=200-50 p+V\), where \(\mathrm{V}\) is a random variable having a probability density defined by

\(f(v)=0.02 I_{[-25,25]}(v)\) and \(p\) is the price of milk, in dollars per gallon.

a. What is the probability that the quantity demanded will be greater than 100,000 gal if price is equal to \(\$ 2\) ? if price is equal to \(\$ 2.25\) ?

b. If the variable cost of supplying \(Q\) amount of milk is given by the cost function \(C(Q)=20 Q^{5}\), define a random variable that represents the daily profit above variable cost from the sale of milk.

c. If price is equal to \(\$ 2\), what is the probability that there will be a positive profit above variable cost on a given day? What if price is set to \(\$ 2.25\) ?

d. Is there any conceptual problem with using the demand function listed above to model quantity demanded if \(p=4\) ? If so, what is it?