(a) The demand for an item produced daily by Tun Production is given by p = 100 - ax where p is the price per unit and x is the quantity demanded. The total cost, C(x) of producing x units of the item is given by C(x) = b + 5x. Tun Production will obtain the maximum profit when its production is 100 units and the profit is RM 500 if it produces 80 units of item.

(i) Write down the expression of total revenue function and total profit function. [2]

(ii) Compute the values of a and b, given that a and b are constants. [4]

(iii) Compute the marginal profit when 90 units are produced and sold. [2]

(iv) Compute the maximum profit of Tun Production. [2]

(b) A driver has to pay an annual road tax and insurance of RM 900. His car does 30 kilometers to one gallon which costs 75 cents per gallon. The car is service every 3 000 kilometers at a cost of RM 20, and depreciation is calculated in cent by multiplying the square of the mileage by 0.01.

(i) If he covers x kilometers in a year, compute an expression for the total cost in travelling x kilometers and the average total cost per kilometer. [7]

(ii) Show that the annual mileage which will minimize the average total cost per kilometer is around 3000 kilometers. [3]