Market research for a certain ice cream company indicates that if the price per cup of ice cream is \(P 50.00\) , the demand will be \(1200\) cups, whereas if the price is increased to \(P 60.00\) , the demand will be \(1100\) cups.
(a) If \(p\) denotes the price per cup (in pesos) and \(x\) denotes the demand for ice cream (in cups), express \(p\) as a linear function of \(x\) .
(b) Determine the revenue function and its domain.
(c) Find the marginal revenue at a production level of \(200\) cups of ice cream. Interpret.
Now, suppose it is known that the total cost of production of \(x\) cups of ice cream is given by ( \(x ) = 3 x + 25,500\) .
(d) Find the value(s) of \(x\) at the break-even points.
(e) Use marginal analysis to approximate the profit from the sale of the \(500 ^ { th} \) cup of ice cream. Compare this value with the exact profit from the sale of the \(500 ^ { th} \) cup.