The quantity, q, of the Chicago Cubs' tickets sold depends on the selling price, p, in dollars. In other words, the quantity sold is a function of price q = f (p). The Chicago Cubs' ticket revenue is a product of the price and quantity of the tickets sold ( R = q * p ). Chicago sold 2745098 tickets in the year of 2012 which generated a revenue of 140 million dollars. According to statistics of quantity and price the Cubs tickets are sold, the Cubs will generate 33614 (33613.53433 to be precise) less tickets sold if the price exceeds 51 dollars. If f ( 51) = 2745098 and f '( 51 ) = -33613.53433) then at what rate will revenue change per dollar increase?
STEP ONE: DEFINE RULES NEEDED
The question is asking us how the product of price and quantity the at which the tickets are sold effect the total revenue. We are given that f ( 51) = 2745098 and f '( 51 ) = -33613.53433). We are also given that ( R = q * p ). In order to find the rate at which the revenue changes we must find the derivative. Revenue is a product of price and quantity of the tickets sold so the best solution to this problem is through the product rule.
Step two: Define functions
Revenue is a function of quantity and price. Quantity is a function of price. We therefore substitute "q" with f (p) in the equation of Revenue. This creates an equation that depends on the product of 2 functions and we can now apply the product rule. The two functions that produce revenue is f (p) and p . As stated previously f (51) = 2745098 and the price of each ticket is 51 dollars. The photos below show how these functions will look as a product of revenue. Now that we have an equation for revenue we can find the rate at which revenue changes using the product rule.
Step three: PRODUCT RULE
The question is asking for the rate at which the revenue changes so we need to find the derivative of R. Now that we have all of our functions defined, we can then plug them into the product rule.