Final answer:
To find the smallest increase in ticket price that will yield $2,197.50 in revenue for the carnival, we must use the concept of price elasticity of demand and calculate the product of ticket price and quantity sold until we reach the desired total revenue.
Step-by-step explanation:
The student's question involves finding the smallest increase in ticket price that will produce $2,197.50 of revenue on a Saturday at a carnival. The carnival initially sells 3,000 tickets at 65 cents each. With every 10 cent increase in price, it is estimated that 70 fewer tickets will be sold. To calculate the smallest price increase, we must use the concept of price elasticity of demand, as it relates to total revenue, which is the product of the price and the quantity of tickets sold.
Let's set up the equation for total revenue (TR): TR = (initial price + increment * number of increases) * (initial quantity - 70 * number of increases). We need to find the number of 10 cent increases where TR is equal to $2,197.50.
By substituting the given values into the equation and systematically increasing the ticket prices by increments of 10 cents while accounting for the corresponding drop in the amount of tickets sold, we will find the smallest ticket price increase that results in the desired revenue.