5. [Maximizing profits] The weekly demand (measured in thousands of gallons) for gasoline at a rural gas station is a random variable X with probability density function fX(u) = ( 5(1 u) 4 , 0 u 1, 0, elsewhere. Let C (in thousands of gallons) denote the capacity of the tank (which is re-filled weekly.) (a) If C = 0.5, what is the probability that the weekly demand for gasoline can be satisfied? Note that if your answer is (say) 0.666..., then, in the long run, the gas station can supply the weekly demand two weeks out of three. (b) What is the minimum value of C required to ensure that the probability that the demand exceeds the supply is no larger than 105 ? (c) Suppose that the owner makes a gross profit of $0.64 for each gallon of gasoline sold. Let Y denote the amount of gasoline sold per week. Express Y as a function of X, the weekly demand for gasoline? (Hint: the owner cannot sell more gasoline each week than the tank can hold!) What is the average weekly gross profit and how does it behave as a function of C, the tank