Problem 82 - Applied Calculus

Question

SOCIAL SCIENCE: Health Club Attendance A study analyzed how the number of visits a person makes to a health club varies with the monthly membership price. It found that the number of visits per year is given approximately by \(v(x)=-0.004 x^{2}+0.56 x+42,\) where \(x\) is the monthly membership price. What monthly price maximizes the number of visits?

Step-by-Step Solution

Answer

The monthly price that maximizes visits is $70.

Step 1: Understand the Function

The function given is a quadratic equation in the form \( v(x) = -0.004x^2 + 0.56x + 42 \). Since the coefficient of \( x^2 \) is negative, this parabola opens downwards, indicating that it has a maximum point.

Step 2: Determine the Vertex Formula

The vertex of a quadratic function \( ax^2 + bx + c \), where \( a eq 0 \), is given by \( x = -\frac{b}{2a} \). This \( x \)-value gives the monthly membership price that maximizes the visits.

Step 3: Substitute and Solve

Using the coefficients from the function \( a = -0.004 \) and \( b = 0.56 \), substitute into the vertex formula: \( x = -\frac{0.56}{2(-0.004)} \).

Step 4: Calculate the Maximum Value

Calculate the value: \( x = \frac{0.56}{0.008} = 70 \). Thus, the monthly price that maximizes the number of visits is \( 70 \).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function characterized by its highest degree term being squared. It follows the general form of \[ ax^2 + bx + c, \]where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic functions graph as a curve called a parabola. In the function provided, \[ v(x) = -0.004x^2 + 0.56x + 42, \]we see that the term \(-0.004x^2\) determines that the parabola opens downwards because its coefficient is negative. This is crucial for identifying that the function will have a maximum point rather than a minimum.

Some key points include:

• The parabola is symmetrical around its vertex, where the maximum or minimum value of the function occurs, depending on the direction the parabola opens.
• Quadratics are used to model various real-world phenomena, from projectile motion to economics.
• In this context, understanding how the number of visits changes with membership price can help in making strategic business decisions.

Vertex Formula

The vertex of a parabola is the point where it turns; it represents the maximum or minimum of the quadratic function. The location of the vertex can be found using the vertex formula, which is particularly useful in quadratic optimization problems. For a quadratic equation \\[ ax^2 + bx + c, \]the vertex \( x \)-coordinate is given by the formula\[ x = -\frac{b}{2a}. \]This formula is derived by completing the square of the quadratic equation or through calculus by setting the derivative to zero to find the stationary point.

In the problem, we applied this formula to our specific function. By substituting \( a = -0.004 \) and \( b = 0.56 \) into the formula, we found the monthly membership price that maximizes health club visits to be 70.

Knowing how to use the vertex formula is essential as it allows:

• Quick determination of crucial points in quadratic models without needing to graph the function.
• Efficient solving of complex real-world problems by identifying optimal values.

Parabola Properties

A parabola has several key properties that make it an interesting and useful graph in mathematics and its applications. Here are some important features:

• Direction: Determined by the sign of \( a \):
  • If \( a > 0 \), the parabola opens upwards and has a minimum point.
  • If \( a < 0 \), the parabola opens downwards and has a maximum point, as is the case in the exercise given.
• Vertex: The vertex is either the highest or lowest point on the graph. For a downward opening parabola, the vertex is the maximum value.
• Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. The equation for the axis of symmetry is \( x = -\frac{b}{2a} \).
• Intercepts:
  • The \( y \)-intercept is the point where the parabola crosses the \( y \)-axis, and it is found by evaluating the function at \( x = 0 \).
  • The \( x \)-intercepts (if they exist) are the points where the graph crosses the \( x \)-axis, found by solving \( ax^2 + bx + c = 0 \).

These properties not only aid in graphing but also in gaining a deeper understanding of how changes in the function's coefficients affect the graph's shape. Recognizing these properties will improve problem-solving skills in quadratic functions and their applications.