Problem 57 - Algebra 2

Question

FUND-RAISING For Exercises 56 and 57 , use the following information. Last year, 300 people attended the Sunnybrook High School Drama Club's winter play. The ticket price was \(\$ 8 .\) The advisor estimates that 20 fewer people would attend for each \(\$ 1\) increase in ticket price. If the Drama Club raised its tickets to this price, how much income should it expect to bring in?

Step-by-Step Solution

Answer

The expected income by raising ticket prices to maximize revenue is $2645.

Step 1: Define Variables

Let's define the variables needed to calculate the expected income. Let \( x \) represent the amount (in dollars) by which the ticket price is increased. The new ticket price will then be \( 8 + x \) dollars. With each \( \$1 \) increase, 20 fewer people will attend, thus the number of attendees will be \( 300 - 20x \).

Step 2: Write the Income Equation

The income, \( I \), from ticket sales is calculated by multiplying the number of attendees by the ticket price. Therefore, the income function based on the increased price is \( I = (8 + x)(300 - 20x) \).

Step 3: Expand the Income Equation

Expand the equation \( I = (8 + x)(300 - 20x) \) to a quadratic form:\[ I = 8(300 - 20x) + x(300 - 20x) \]\[ = 2400 - 160x + 300x - 20x^2 \]Combine like terms:\[ I = -20x^2 + 140x + 2400 \]

Step 4: Find Maximum Income

The income equation \( I = -20x^2 + 140x + 2400 \) is a quadratic equation in the standard form \( ax^2 + bx + c \). Its graph is a parabola that opens downwards because the coefficient of \( x^2 \) is negative. To find the price that maximizes income, calculate the vertex \( x \) using \( x = -\frac{b}{2a} \):\[ x = -\frac{140}{2(-20)} = 3.5 \]

Step 5: Calculate Maximum Income

Substitute \( x = 3.5 \) back into the income equation to find the maximum income:\[I = -20(3.5)^2 + 140(3.5) + 2400 \]Calculate each term:\[I = -20(12.25) + 490 + 2400 \]\[I = -245 + 490 + 2400 \]\[I = 2645 \]

Step 6: Answer the Question

The maximum income that the Drama Club can expect by raising the ticket price by \\(3.5 (making the tickets \\)11.5 each) is \$2645.

Key Concepts

Variables Definition

When tackling problems involving quadratic equations, defining variables clearly makes solving them much easier. In our scenario, we start by deciding what we need to find. We want to know how changes in ticket pricing will affect the number of attendees and income.

• Let's take the variable \( x \) to represent the increase in ticket price in dollars. This choice helps us link price changes directly to the equation.
• The adjusted ticket price becomes \( 8 + x \), since the original ticket price is \$8.
• Attendance drops by 20 people for each dollar increase. Thus, the number of tickets sold will be \( 300 - 20x \).

By establishing these variables, we lay a foundation for calculating the income generated from ticket sales, allowing us to construct a viable income function.

Income Calculation

The income from ticket sales can be seen as a product of two factors: the number of tickets sold and the ticket price. Once we have our variables defined, we can establish this relationship in a mathematical equation.

The income function, \( I \), is defined as:\[I = (8 + x)(300 - 20x)\]This equation essentially states:

• The first term \( (8 + x) \) is the new ticket price, incorporating any increase \( x \).
• The second term \( (300 - 20x) \) calculates how many tickets are sold based on this new price.

By multiplying these, we determine the total income. The outcome is influenced by both how many people attend due to price changes and the new revenue per ticket.

Vertex of a Parabola

In the context of quadratic equations, the vertex plays an important role in optimization problems. Here, it helps us find the peak or maximum value of our income function. Quadratic equations take the form \( ax^2 + bx + c \); for our income equation:\[I = -20x^2 + 140x + 2400\]The equation is in the standard quadratic form, where:

• \( a = -20 \)
• \( b = 140 \)
• \( c = 2400 \)

A parabola opens downward if the coefficient \( a \) is negative. The vertex's x-coordinate, which gives the point of maximum income, is found using the formula: \[ x = -\frac{b}{2a} \]Substituting our values:\[ x = -\frac{140}{2(-20)} = 3.5 \]This step is essential for identifying where income is maximized.

Maximum Income

By finding the vertex, we know the optimal price increase for maximizing income. Let's use this result to compute the maximum income attainable. Given that our vertex occurs at \( x = 3.5 \), follow with substitution into our income equation:\[I = -20(3.5)^2 + 140(3.5) + 2400\]Breaking it down:

• Calculate \(-20(3.5)^2 = -245\)
• Compute \(140(3.5) = 490\)
• Addition gives \( -245 + 490 + 2400 = 2645\)

This income, \\(2645, represents the maximum achievable under the conditions defined. Pricing tickets at \\)11.5 optimizes revenue given the model parameters of decreasing attendance. Appreciating this step restfully depicts how mathematical modeling connects to real-world financial strategy.