Question

You throw a basketball whose path can be modeled by \(y=-16 x^{2}+15 x+6,\) where \(x\) represents time (in seconds) and \(y\) represents height of the basketball (in feet). a. What is the maximum height that the basketball reaches? b. In how many seconds will the basketball hit the ground if no one catches it?

Short answer

The basketball reaches its maximum height of about 7.734375 feet. The basketball will hit the ground after approximately 0.867024 and 6.88298 seconds, but we only consider the higher value since the basketball starts from the ground. So, it will hit the ground after about 6.88298 seconds.

Step-by-step solution

Find the vertex of the parabola

The equation for the x-coordinate of the vertex of a parabola given by \(y = ax^2 + bx + c\) is \(x = -\frac{b}{2a}\). Apply this with \(a = -16\) and \(b = 15\) to find the time at which the basketball reaches its maximum height. That is, \(x = -\frac{15}{2*(-16)}\).

Calculate the maximum height

Substitute the x-coordinate of the vertex into the original equation to find the y-coordinate, which represents the maximum height of the basketball. That is, \(y = -16 x^{2} + 15x + 6\).

Find the time when the basketball hits the ground

Set the equation equal to zero and solve for \(x\). The quadratic formula will need to be used as this equation does not factor easily. The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Since time cannot be negative, choose the positive root.

Key concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial when dealing with quadratic functions, especially if you're trying to find the maximum or minimum values. The vertex of a parabola is the point where the curve changes direction, and it represents either the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.

In the given exercise, the basketball's path follows a downward opening parabola due to the negative coefficient of the squared term. To find the vertex, you can use the formula for the x-coordinate which is \( x = -\frac{b}{2a} \). This concise method avoids the need for completing the square and provides an efficient way to determine the time at which the basketball reaches its peak height.

Maximum Height of a Projectile

When you throw an object, like a basketball, it follows a path known as a projectile motion. The maximum height of a projectile is an important characteristic of this path and is represented by the highest point on the parabola's curve — the vertex. The y-coordinate of the vertex gives you this maximum height.

Using the coordinates of the vertex obtained earlier, specifically the x-value which is time, we can determine the y-value, or the maximum height, by substituting it back into the original quadratic equation. In our case, the formula \( y = -16x^{2} + 15x + 6 \) provides the height at any given time, so replacing the x with the x-coordinate of the vertex will give the peak height during the basketball's flight.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation, where roots are the values for which the quadratic equation equals zero. This formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where the ± symbol indicates that you will usually get two solutions for x. These solutions correspond to the x-intercepts of the parabola.

It's particularly useful when the quadratic doesn't factor nicely or when simplifying by other methods is too cumbersome. In solving for the time when the basketball hits the ground, since it's a real-world problem, we disregard the negative solution because negative time doesn't make sense in the given context.

Solving Quadratic Equations

Quadratic equations come in the standard form \( ax^2 + bx + c = 0 \) and can be solved using various methods, including factoring, completing the square, using the quadratic formula, or graphing. Factoring involves rewriting the quadratic as a product of binomials, while completing the square transforms the equation into a perfect square trinomial.

However, not all quadratic equations can be easily factored. In those instances, the quadratic formula is an essential method. As we've seen in the basketball example, it allowed us to find the precise moment when the basketball would hit the ground by setting the height to zero and solving for the positive root since only positive time values are relevant here.