Final answer:
To find the optimal ticket price for the Wildcat Theatre, we develop a revenue function based on the initial conditions of 300 moviegoers and $8.50 per ticket. We then calculate the vertex of the quadratic function representing revenue as a function of the number of fifty-cent price increases. The minimum sales occur when the ticket price increases to the point that the number of moviegoers drops to zero.
Step-by-step explanation:
To determine the ticket price the Wildcat Theatre should charge to maximize their ticket sales, we need to develop a revenue function, R(x), where 'R' is the total revenue and 'x' is the number of fifty-cent price increases. Considering the theatre starts with 300 moviegoers at a price of $8.50, we can express the average number of moviegoers as a function of 'x' as follows: 300 - 4x. Because for every fifty-cent increase we lose 4 moviegoers, the new price after 'x' increases would be $8.50 + 0.50x. So the revenue function can be modelled as R(x) = (300 - 4x)($8.50 + 0.50x). To maximize revenue, we need to find the vertex of this quadratic function, which represents the maximum point.
The vertex occurs at -b/2a when a function is expressed in standard quadratic form. So, if we expand the revenue function and write it in standard form, we can find 'a' and 'b' and hence the vertex. This point will give us the optimal number of fifty-cent increases 'x', which we can use to determine the best ticket price and the corresponding number of moviegoers for maximum revenues.
To ascertain the minimum sales, we assume the trend of decreasing moviegoers continues beyond the point of revenue maximization until the revenue is zero (i.e., no one goes to the movies anymore). From the expanded revenue function, we set R(x) to zero and solve for 'x' to see where that would be.