Final answer:
The demand function, p(x), given the number of spectators, x, is p(x) = -2/7000 x + 44/7. To maximize revenue, the ticket price should be set at the point where the product of price and quantity of tickets sold is maximized, requiring consideration of the price elasticity of demand.
Step-by-step explanation:
To find the demand function p(x), where x is the number of spectators, we can use the given points (24000, $12) and (31000, $10) to form a linear function. First, find the slope m of the line using Δy/Δx = (10 - 12) / (31000 - 24000) = -2/7000. Then use the point-slope form to create the equation of the line: p(x) - y1 = m(x - x1), which gives us p(x) - 12 = (-2/7000)(x - 24000).
Simplifying this equation, we get the demand function p(x) = -2/7000 x + (12 + 2/7000*24000), which simplifies to p(x) = -2/7000 x + 44/7.
To maximize revenue, we need to find the price that results in the maximum product of price and quantity of tickets sold. This is often near the midpoint of the linear demand curve, where the price elasticity of demand indicates a balance between the price and the quantity sold. The maximum revenue is not necessarily at the highest price because the number of tickets sold may decrease too much, nor at the lowest price where quantity sold isn't enough to compensate for the low price.