Question

Tickets to a school dance cost two dollars in the projected attendance is 450 people it is further projected that for every $.10 increase in ticket price the average attendance will decrease by six at what ticket price of the receipts from the dance be 1050

Answer 1

To solve this problem, let's break it down step by step.

First, let's denote the original ticket price as \(P\) dollars. The total revenue from ticket sales is given by the formula:

\[ \text{Revenue} = \text{Ticket Price} \times \text{Number of People} \]

In this case, the ticket price is \(P\) dollars, and the number of people attending the dance is 450.

So, the revenue is \(450P\) dollars.

According to the projection, for every \(0.10\) dollar increase in ticket price, the average attendance will decrease by 6 people. This implies that the new attendance, \(N\), is related to the original attendance by the equation:

\[ N = 450 - 6 \times \left( \frac{P - 2}{0.10} \right) \]

The receipts from the dance would be the product of the new ticket price and the new attendance, which is \(N \times (P + 0.10)\).

We want the receipts to be \(1050\) dollars. Substituting the expression for \(N\) from the equation above, we get:

\[ 1050 = \left( 450 - 6 \times \left( \frac{P - 2}{0.10} \right) \right) \times (P + 0.10) \]

Now, you can solve this equation for \(P\) to find the ticket price at which the receipts from the dance will be \(1050\) dollars. Please note that this is a quadratic equation, so you may need to use a calculator or software to find the exact solution.

Answer 2

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Explanation:

Let's solve this problem step by step.

Let's assume the initial ticket price is $2 and the initial attendance is 450 people.

Now, we need to determine how many $0.10 increases in ticket price will lead to a decrease in attendance by 6 people.

For every $0.10 increase, the attendance decreases by 6 people.

So, for every decrease of 6 people, there is an increase of $0.10 in ticket price.

To find the total decrease in attendance, we divide the desired decrease in receipts ($1050) by $0.10:

Total decrease in attendance = $1050 / $0.10 = 10500 / 1 = 10500

Now, we divide the total decrease in attendance (10500) by the decrease in attendance per $0.10 increase (6):

Number of $0.10 increases = 10500 / 6 = 1750 / 1 = 1750

To find the final ticket price, we multiply the number of $0.10 increases by $0.10 and add it to the initial ticket price:

Final ticket price = ($2 + 1750 * $0.10) = ($2 + $175) = $177

Therefore, the ticket price at which the receipts from the dance will be $1050 is $177.