Problem 46 - Beginning and Intermediate Algebra

Question

Professor Barbu has found that the number of students attending his intermediate algebra class is approximated by$$S(x)=-x^{2}+20 x+80$$ where \(x\) is the number of hours that the Campus Center is open daily. Find the number of hours that the center should be open so that the number of students attending class is a maximum. What is this maximum number of students?

Step-by-Step Solution

Answer

The center should be open for 10 hours. The maximum number of students is 180.

Step 1: Identify the quadratic equation

The given quadratic equation for the number of students is \( S(x) = -x^2 + 20x + 80 \). This equation is in the form \( ax^2 + bx + c \) where \( a = -1 \), \( b = 20 \), and \( c = 80 \).

Step 2: Determine the vertex of the parabola

The maximum value of a quadratic function \( ax^2 + bx + c \) occurs at the vertex of the parabola. The vertex \( x \) can be found using the formula \( x = \frac{-b}{2a} \).

Step 3: Calculate the x-coordinate of the vertex

Substitute \( a = -1 \) and \( b = 20 \) into the formula: \[ x = \frac{-20}{2(-1)} = \frac{-20}{-2} = 10 \] So, the number of hours the Campus Center should be open for maximum student attendance is 10 hours.

Step 4: Find the maximum number of students

To find the maximum number of students, substitute \( x = 10 \) back into the original equation: \( S(10) = -10^2 + 20(10) + 80 \).

Step 5: Evaluate the function at \( x = 10 \)

Calculate the value: \[ S(10) = -100 + 200 + 80 = 180 \] Therefore, the maximum number of students attending the class is 180.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the form: ewline ax² + bx + c = 0 where,

• a is the coefficient of x²
• b is the coefficient of x
• c is the constant term

In our example, the equation that models the number of students attending Professor Barbu's class is: \[ S(x) = -x² + 20x + 80 \]This means:

• a = -1
• b = 20
• c = 80

Quadratic equations typically represent parabolas, which are U-shaped curves. The parabola can either open upwards (like a U) or downwards (like an inverted U). In this case, because the coefficient of x² is negative (a = -1), our parabola opens downwards. This indicates that the vertex of the parabola will give us the maximum value for our function.

Using the Vertex Formula

The vertex of a parabola gives us the maximum or minimum point of a quadratic function. When a parabola opens downwards, as in our example, the vertex will represent the maximum point. The vertex can be found using the vertex formula: \[ x = \frac{{-b}}{{2a}} \] Let's calculate the x-coordinate of the vertex: a = -1 and b = 20 ewline Substitute these into the equation: \[ x = \frac{{-20}}{{2(-1)}} = \frac{{-20}}{{-2}} = 10 \] So, the Campus Center should be open for 10 hours to maximize student attendance. ewline To find the maximum number of students, substitute x = 10 into the original quadratic function: \[ S(10) = -10² + 20(10) + 80 \] This simplifies to: \[ S(10) = -100 + 200 + 80 = 180 \] Thus, the maximum number of students attending is 180.

Graphing Parabolas and Finding the Vertex

A parabola is a symmetric curve, and its highest or lowest point is called the vertex.

• For the function S(x) = -x² + 20x + 80, its graph is a parabola opening downwards.
• The vertex of this parabola gives the maximum number of students.

To visualize this, imagine the parabola on a graph:

1. As x increases or decreases from 10, the number of students (value of S(x)) will decrease since the parabola opens downwards.
2. To draw this parabola, plot several points by substituting different values of x into the quadratic function.
3. The vertex is at (10, 180), meaning 10 hours of the Campus Center being open results in a maximum of 180 students attending.

By plotting other points, you'll see the shape of the graph and understand why 10 hours results in the maximum number of students. This visual representation can help solidify your understanding of how quadratic functions and their vertices work.