An auditorium has 1600 seats. At a ticket price of $6, the owners can sell 1200 tickets. For each 25 cent increase in price the tickets sold reduces by 25. What price should the owners set to maximize the income?
Income=(6+.25x)(1200-25x)
Generalize and analyze this equation.
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To solve this problem, we need to find the price that maximizes the income. The income is given by the equation Income = (6 + 0.25x)(1200 - 25x), where x represents the number of 25 cent increases in ticket price.
To analyze the equation and find the value of x that maximizes the income, we need to find the derivative of the income function with respect to x and set it equal to zero. This will give us the critical point where the slope of the income function is zero.
Let's go through the process step by step:
1. Expand the income equation: Income = (6 + 0.25x)(1200 - 25x)
Income = 7200 + 1800x - 150x^2 - 6x
2. Find the derivative of the income function with respect to x:
d(Income)/dx = 1800 - 300x - 6
3. Set the derivative equal to zero and solve for x:
1800 - 300x - 6 = 0
300x = 1794
x = 1794/300
x = 5.98
4. We need to round x to the nearest whole number, as we can't have a fraction number of increases. So x = 6.
5. Substitute the value of x back into the income equation to find the corresponding price:
Price = 6 + 0.25(6)
Price = 6 + 1.50
Price = $7.50
Therefore, to maximize the income, the owners should set the ticket price at $7.50.
Note: The analysis assumes that there is linear correlation between the increase in ticket price and the decrease in number of tickets sold. This might not always hold true, as there could be other factors at play.