Answer:
$35 or $40
Explanation:
To find the ticket price that will generate the maximum revenue, we need to consider the relationship between the ticket price, the attendance, and the revenue.
Let's start by defining some variables:
- Let "P" be the ticket price in dollars.
- Let "A" be the attendance (number of people).
We know that the ticket price is initially $25, so P = $25, and the projected attendance is 500 people, so A = 500.
The revenue (R) is calculated by multiplying the ticket price by the attendance:
R = P · A
Now, we have to consider the impact of increasing the ticket price by $5. For every $5.00 increase in the ticket price, the dance committee projects that the attendance will decrease by 50.
So, if the ticket price is increased by $5, the new price will be P + $5, and the new attendance will be A - 50.
Now, the new revenue (R') with the increased ticket price can be calculated:
∴ R' = (P + $5) · (A - 50)
To find the value of the maximum revenue we will use the formulas we have found and plug in numbers (ticket price and attendance) until we find the largest revenue.
In case 1:
⇒ R = P · A
⇒ R₀ = (25)(500) = $12,500
In case 2:
⇒ R' = (P + $5) · (A - 50)
⇒ R₁ = ($25 + $5) · (500 - 50) = ($30)(450) = $13,500
In case 3:
⇒ R₂ = ($30 + $5) · (450 - 50) = ($35)(400) = $14,000
In case 4:
⇒ R₃ = ($35 + $5) · (400 - 50) = ($40)(350) = $14,000
In case 5:
⇒ R₄ = ($40 + $5) · (350 - 50) = ($45)(300) = $13,500
In case 6:
⇒ R₅ = ($45 + $5) · (300 - 50) = ($50)(250) = $12,500
If you kept going, the revenue will continue to decrease. Thus, we can conclude the dance will maximize revenue with either a ticket price of $35 or $40.
Additional Information:
Revenue: Revenue is the total income generated by selling a certain number of products or services at a given price. In this context, revenue is obtained by multiplying the ticket price by the number of attendees. Maximizing revenue is a common goal for businesses and organizations to optimize their income.