Question
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). What does the number 24 represent in the expression?
Answers
Answered by
GPT-4o mini
In the expression \( p(840 - 35p) \) for total revenue, where \( p \) is the price per ticket, we can interpret the components of the equation to better understand what the number 24 signifies.
The revenue model can be viewed as a quadratic function in terms of \( p \):
\[
R(p) = p(840 - 35p) = 840p - 35p^2
\]
This function opens downwards (as indicated by the negative coefficient of the \( p^2 \) term), suggesting that there is a maximum revenue achieved at a certain ticket price.
Factoring the expression gives us:
\[
R(p) = -35p(p - 24)
\]
From this factored form, we see that the expression has roots at \( p = 0 \) and \( p = 24 \). The root at \( p = 0 \) indicates that if the price of the ticket is zero, no revenue is generated because no tickets are sold. The root at \( p = 24 \) is significant because it indicates the price point at which if tickets were sold for $24, the revenue would also be zero (i.e., no sales).
However, more importantly, in the context of the quadratic revenue equation, the number 24 represents the ticket price at which the total revenue reaches its maximum. The vertex of the parabola defined by the revenue function occurs at this point. To find the maximum revenue price, we can use the vertex formula \( p = -\frac{b}{2a} \), where \( R(p) = ap^2 + bp + c \):
Here \( a = -35 \) and \( b = 840 \):
\[
p = -\frac{840}{2 \cdot -35} = \frac{840}{70} = 12
\]
However, it appears there was an inconsistency; let’s state that:
The maximum revenue occurs at a price value of \( p = 12 \) and we have the factor \( (p-24) \). This shows the symmetry of the quadratic. The vertex would be the maximum revenue, hence:
The number 24 indicates the price point at which revenue would become zero again when the price is too high—indicating a 'break-even' point, after which further price increases lead to no sales, thus giving a meaningful notion of demand constraints.
In conclusion, the number \( 24 \) in the expression essentially represents the ticket price at which the model indicates zero revenue.
The revenue model can be viewed as a quadratic function in terms of \( p \):
\[
R(p) = p(840 - 35p) = 840p - 35p^2
\]
This function opens downwards (as indicated by the negative coefficient of the \( p^2 \) term), suggesting that there is a maximum revenue achieved at a certain ticket price.
Factoring the expression gives us:
\[
R(p) = -35p(p - 24)
\]
From this factored form, we see that the expression has roots at \( p = 0 \) and \( p = 24 \). The root at \( p = 0 \) indicates that if the price of the ticket is zero, no revenue is generated because no tickets are sold. The root at \( p = 24 \) is significant because it indicates the price point at which if tickets were sold for $24, the revenue would also be zero (i.e., no sales).
However, more importantly, in the context of the quadratic revenue equation, the number 24 represents the ticket price at which the total revenue reaches its maximum. The vertex of the parabola defined by the revenue function occurs at this point. To find the maximum revenue price, we can use the vertex formula \( p = -\frac{b}{2a} \), where \( R(p) = ap^2 + bp + c \):
Here \( a = -35 \) and \( b = 840 \):
\[
p = -\frac{840}{2 \cdot -35} = \frac{840}{70} = 12
\]
However, it appears there was an inconsistency; let’s state that:
The maximum revenue occurs at a price value of \( p = 12 \) and we have the factor \( (p-24) \). This shows the symmetry of the quadratic. The vertex would be the maximum revenue, hence:
The number 24 indicates the price point at which revenue would become zero again when the price is too high—indicating a 'break-even' point, after which further price increases lead to no sales, thus giving a meaningful notion of demand constraints.
In conclusion, the number \( 24 \) in the expression essentially represents the ticket price at which the model indicates zero revenue.