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The Beech Grove town council is organizing this year's downtown fair. The council members have found that when they charge more for each entrance ticket, they sell fewer tickets overall. The fair's total revenue from selling entrance tickets, in dollars, can be modeled by the expression p(840−35p)p(840 - 35p), where pp is the price per ticket in dollars. This expression can be written in factored form as −35p(p−24)-35p(p - 24).\newlineWhat does the number 2424 represent in the expression?\newline(A)the fair's minimum total revenue in dollars\newline(B)the price per entrance ticket in dollars that maximizes the fair's total revenue\newline(C)the price per entrance ticket in dollars so that the fair's total revenue is zero\newline(D)the fair's maximum total revenue in dollars

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Q. The Beech Grove town council is organizing this year's downtown fair. The council members have found that when they charge more for each entrance ticket, they sell fewer tickets overall. The fair's total revenue from selling entrance tickets, in dollars, can be modeled by the expression p(840−35p)p(840 - 35p), where pp is the price per ticket in dollars. This expression can be written in factored form as −35p(p−24)-35p(p - 24).\newlineWhat does the number 2424 represent in the expression?\newline(A)the fair's minimum total revenue in dollars\newline(B)the price per entrance ticket in dollars that maximizes the fair's total revenue\newline(C)the price per entrance ticket in dollars so that the fair's total revenue is zero\newline(D)the fair's maximum total revenue in dollars
  1. Revenue Expression Analysis: The expression for total revenue is p(840−35p)p(840 - 35p). To understand what 2424 represents, we need to look at the factored form −35p(p−24)-35p(p - 24).
  2. Finding Root at p=24p = 24: In the factored form, the expression is zero when p=0p = 0 or p=24p = 24. Since p=0p = 0 is trivial (price can't be zero), p=24p = 24 is the other root of the equation.
  3. Interpreting Root Value: The root p=24p = 24 represents the price at which the total revenue changes direction; it's the vertex of the parabola representing the revenue function. This is the price that maximizes total revenue.
  4. Maximizing Total Revenue: Therefore, the number 2424 represents the price per entrance ticket in dollars that maximizes the fair's total revenue.

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