Problem 44 - College Algebra

Question

A football is thrown by a quarterback to a receiver 40 yards away. The quadratic function $$s(t)=-0.025 t^{2}+t+5$$ models the football's height above the ground, s(t), in feet, when it is \(t\) yards from the quarterback. How many yards from the quarterback does the football reach its greatest height? What is that height?

Step-by-Step Solution

Answer

The football reaches its greatest height of 15 feet when it is 20 yards from the quarterback.

Step 1: Finding the distance

Use the formula \(-\frac{b}{2a}\) to find the x-coordinate of the vertex, which is the distance from the quarterback. With \(a = -0.025\) and \(b = 1\), we get the distance \(t = -\frac{1}{2(-0.025)} = 20\) yards.

Step 2: Finding the height

Substitute this value into the equation to find the y-coordinate of the vertex, which represents the greatest height reached by the ball. We get \(s(20) = -0.025(20)^2 + 20 + 5 = 15\) feet.

Key Concepts

Understanding the Vertex Formula

When dealing with quadratic functions, the vertex formula is an essential tool. It helps us determine the vertex of a parabola. In mathematical terms, the vertex of a quadratic function given in standard form \[ ax^2 + bx + c \] has its x-coordinate determined by the formula \[ x = -\frac{b}{2a} \].The vertex is a crucial point because it represents the maximum or minimum value of the quadratic function, depending on whether the parabola opens upwards or downwards. The value of \(a\) in the quadratic function indicates the parabola's opening direction:

• If \(a > 0\), the parabola opens upward, and the vertex is a minimum point.
• If \(a < 0\), the parabola opens downward, and the vertex is a maximum point.

Understanding this allows us to solve real-world problems, such as finding the maximum height of a thrown football, as in our given exercise.

Calculating the Maximum Height

Once we have found the x-coordinate of the vertex using the vertex formula, we can easily determine the corresponding maximum height of the parabola. For the quadratic equation representing the football's flight:\[ s(t) = -0.025 t^{2} + t + 5 \]We substitute the x-coordinate of the vertex, found in the previous step, into the equation to find the value of the function at this point. This y-coordinate is the maximum height reached by the object.In this scenario, the equation tells us about the behavior of the football:

• With the vertex at \( t = 20 \) yards, substituting \( t \) in the equation gives \( s(20) = -0.025(20)^2 + 20 + 5 \).
• Simplifying, we find that the maximum height is 15 feet.

This approach illustrates how the mathematics of quadratics extends into tangible situations like tracking the peak height of a football during a pass.

Grasping Quadratic Equations

Quadratic equations are polynomial equations of degree two, generally expressed in the form \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). These equations create parabolas when graphed and can open either upwards or downwards. A quadratic equation helps in calculating various real-world scenarios by modeling paths, speeds, profits, or heights.Key characteristics of quadratic equations include:

• The coefficient \(a\) influences the direction and "width" of the parabola.
• The term \(b\) affects the parabola's axis of symmetry.
• \(c\) indicates the y-intercept, the point where the graph intersects the y-axis.

By understanding these characteristics, you can better solve quadratic equations and apply them to problems such as optimizing the trajectory of a projectile or determining the optimum dimensions for maximum reach.