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 $p(840 - 35p)$ , where $p$ is the price per ticket in dollars. This expression can be written in factored form as $-35p(p - 24)$ . $\newline$ What does the number $24$ 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)

, where

p

is the price per ticket in dollars. This expression can be written in factored form as

-35p(p - 24)

\newline

What does the number

24

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

Revenue Expression Analysis: The expression for total revenue is $p(840 - 35p)$ . To understand what $24$ represents, we need to look at the factored form $-35p(p - 24)$ .
Finding Root at $p = 24$ : In the factored form, the expression is zero when $p = 0$ or $p = 24$ . Since $p = 0$ is trivial (price can't be zero), $p = 24$ is the other root of the equation.
Interpreting Root Value: The root $p = 24$ 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.
Maximizing Total Revenue: Therefore, the number $24$ represents the price per entrance ticket in dollars that maximizes the fair's total revenue.