Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

The path of a diver is approximated by \(y=-\frac{4}{9} x^{2}+\frac{24}{9} x+12\) where \(y\) is the height (in feet) and \(x\) is the horizontal distance (in feet) from the end of the diving board (see figure). What is the maximum height of the diver?

Step-by-Step Solution

Answer

The maximum height of the diver is 16 feet.

Step 1: Understanding the Form of the Function

The function \(y=-\frac{4}{9} x^{2} + \frac{24}{9}x + 12\) is in the form of \(y = ax^2 + bx + c\), which is a quadratic equation. Since the coefficient of \(x^2\) is negative, this will be a concave parabola. The maximum or minimum value of a quadratic function is at its vertex.

Step 2: Finding the Vertex of the Parabola

The x-coordinate of the vertex of a parabola given in standard form is calculated by the formula \(-\frac{b}{2a}\). Substituting the values of 'a' and 'b' from our equation, the 'x' value at the vertex will be \(-\frac{b}{2a} = -\frac{24/9}{2*(-4/9)} = 3\).

Step 3: Finding the Maximum Height of the Diver

Substitute 'x' = 3 (found in step 2) into the quadratic equation to get the 'y' value. \(y = -\frac{4}{9} * 3^{2} + \frac{24}{9} * 3 + 12 = 16\)

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial concept in many mathematical problems, especially when determining the maximum or minimum values of a quadratic function. The path of a projectile, such as a diver's jump or a ball thrown into the air, can be modeled by a quadratic equation, forming a parabola when plotted. The vertex represents the peak in case of a concave down parabola or the lowest point for a concave up parabola. As the name suggests, it's the turning point where the parabola changes direction.

To locate the vertex, we use the formula \( -\frac{b}{2a} \) where 'a' and 'b' are coefficients from the quadratic equation in standard form \( y = ax^2 + bx + c \). In the context of our diving example, the calculation reveals that when the horizontal distance from the board is 3 feet, the diver reaches the peak of the dive. The corresponding 'y' value at this 'x' gives us the maximum height reached by the diver.

Quadratic Function

A quadratic function is represented by the equation \( y = ax^2 + bx + c \), where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola. The sign of the coefficient 'a' determines whether the parabola opens upwards (if 'a' is positive) or downwards (if 'a' is negative), which in turn affects whether the vertex is a minimum or maximum point.

In our diver's example, the negative 'a' value indicates that the parabola opens downwards, meaning the vertex of the parabola will be at the maximum height the diver achieves. By understanding this concept, students can apply the standard form of a quadratic equation not only to estimate the height of jumps but also to problems involving areas, economics, and physics where quadratic functions come into play.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is \( y = ax^2 + bx + c \), where 'y' is the output variable, 'x' is the input variable or independent variable, and 'a', 'b', and 'c' are constants that determine the shape, position, and orientation of the parabola. 'a' cannot be zero, as there would no longer be a square term to make the equation quadratic.

The 'c' term is the y-intercept of the parabola, which is the point where the graph crosses the y-axis. In our diver example, the number 12 indicates where the diver begins in terms of height when at zero distance from the board. This form is very useful as it gives us a straightforward way to plot the quadratic graph and analyze its properties such as the axis of symmetry, intercepts, and the vertex.

Parabolic Motion

Parabolic motion describes the path of an object under the influence of gravity, assuming there's no air resistance. It follows the shape of a parabola, a curve that represents the trajectory of the object. In real-world scenarios such as a diver jumping off a board or a basketball arc after being thrown, the motion can be modeled by a quadratic equation.

Mathematically, this motion is dictated by the second-degree polynomial in our quadratic function, where the squared term representing horizontal distance is modulated by the gravitational pull acting vertically down. As seen in the example of the diver, analyzing the quadratic equation can help us calculate optimal conditions such as maximum height, which is essential in sports strategy, engineering, and even entertainment industry stunt planning.