Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

The growth rate \(y\) (in pounds per month) of an infant is related to present weight \(x\) (in pounds) by the formula \(y=c x(21-x)\), where \(c\) is a positive constant and \(0

Step-by-Step Solution

Answer

The maximum growth rate occurs when the infant weighs 10.5 pounds.

Step 1: Understand the Equation

The formula given is \( y = c x (21 - x) \). This represents a quadratic equation in the form of \( ax^2 + bx + c \) where the coefficient \( a \) will be negative, making it a downward opening parabola. Our task is to find the value of \( x \) that gives the maximum \( y \), which corresponds to the vertex of the parabola.

Step 2: Identify the Vertex

To find the vertex of the parabola described by the equation \( y = c x (21 - x) \), rewrite it in the standard quadratic form: \( y = -c x^2 + 21c x \). The \( x \)-coordinate of the vertex of a quadratic equation \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).

Step 3: Apply the Formula for the x-coordinate of the Vertex

In the equation \( y = -c x^2 + 21c x \), identify \( a = -c \) and \( b = 21c \). Plug these into the formula \( x = -\frac{b}{2a} \): \[ x = -\frac{21c}{2(-c)} = \frac{21}{2} \]

Step 4: Simplify and Interpret

From the calculation in Step 3, we have \( x = \frac{21}{2} = 10.5 \). This is the weight in pounds at which the maximum growth rate occurs, given that \( 0 < x < 21 \).

Step 5: Verify Constraints

Finally, verify that \( x = 10.5 \) falls within the given constraint \( 0 < x < 21 \). Since it does, this solution is valid.

Key Concepts

Maximization

Maximization refers to the process of finding the highest point, or maximum value, in a set of values. When dealing with quadratic functions, this typically involves determining the peak or highest point on the graph, known as the vertex. In contexts like economics or biology, this can translate to finding the most efficient point, such as the highest profit or maximum growth rate. In the given problem, the growth rate function is a quadratic expression—meaning it resembles a parabola, which either opens upwards or downwards. An upside-down or downward-opening parabola has a single highest point, which is the vertex.
For maximization in such cases, our task is to determine this vertex because it represents the maximum value of the quadratic function. Understanding this is crucial as it allows us to solve practical problems involving maximization, like determining the optimal weight for maximum infant growth rate as mentioned in the exercise.

Vertex Formula

The vertex formula is an essential tool for finding the maximum or minimum value of a quadratic function. For a quadratic equation expressed in standard form,

• The general format is:
  • \( y = ax^2 + bx + c \)

The "vertex formula" helps to pinpoint the vertex of the parabola described by this equation. To find the x-coordinate of the vertex, you use the formula:

• \( x = -\frac{b}{2a} \)

In our exercise, where the quadratic is

• \( y = -c x^2 + 21c x \), we identify the components as:
  • \( a = -c \)
  • \( b = 21c \)

Plugging these into the vertex formula gives us the x-coordinate of the vertex, or the weight at which the maximum growth rate occurs. This step translates the problem from an algebraic expression to a meaningful solution.

Parabola

A parabola is the U-shaped curve you often encounter in quadratic functions. Its specific orientation (opening upwards or downwards) depends on the sign of the coefficient "a" in the quadratic equation

• \( y = ax^2 + bx + c \)

If "a" is positive, the parabola opens upwards and the lowest point is the vertex. If "a" is negative, as it is in our exercise with

• \( a = -c \)

, then the parabola opens downwards, making the vertex the highest point. In this way, the parabola visually represents the concept of maximization within the context of our problem, which helps us identify and interpret the maximum growth rate. This clear visualization as an upside-down curve directly links to the problem's solution, indicating where the peak occurs at its highest.