Problem 59 - College Algebra

Question

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(f(x)\), in feet, can be modeled by $$ f(x)--0.8 x^{2}+2.4 x+6 $$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the function that models the ball's parabolic path.

Step-by-Step Solution

Answer

a. The maximum height of the ball is 7.2 feet, which occurs 1.5 feet from where it was thrown. b. The ball travels approximately 4.2 feet horizontally before it hits the ground. c. The graph is a parabolic curve that reaches a maximum at (1.5, 7.2) and intercepts the x-axis at 0 and 4.2.

Step 1: Determine the maximum height of the ball

Firstly, figure out the maximum height that the ball will reach. As that will occur at the vertex of our parabolic function, we will calculate the \(x\) value of that vertex. For a quadratic function of the form \( ax^{2}+bx+c \), the \(x\) coordinate of the vertex is given by \( -b / 2a \). Plug the coefficients from our function into this formula to get: \( -(2.4) / (2 * (-0.8)) = 1.5\). Now that we know \(x = 1.5\) at the maximum height, just input this value into the function to obtain the maximum height: \(f(1.5) = -0.8 * (1.5)^{2} +2.4 * 1.5 +6 = 7.2\) feet.

Step 2: Calculate the distance the ball travels horizontally

For the second part of the problem, we need to establish when the ball hits the ground, which corresponds to the situation when \(f(x) = 0\). In other words, to find the roots of our quadratic function. Using the quadratic formula, given by \(x = (-b ± sqrt(b^2 - 4ac))/2a\), and substituting in -0.8 for 'a', 2.4 for 'b', and 6 for 'c', we isolate the roots. We only consider the positive root, as the negative root would imply negative distance, which does not make sense in this context. Upon calculating, \(x = 4.2\) feet, so the ball will have moved 4.2 feet horizontally when it hits the ground.

Step 3: Create the graph for the function

Keep in mind, the provided equation describes a parabola, which is symmetrical. Hence, the maximum point (from part a) will be the peak of the parabola, and the roots (solution from part b) will be where the parabola intersects the x-axis. This resulting graph shows a typical projectile motion.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a critical point of a quadratic function, and it designates the maximum or minimum value of the function. In the context of projectile motion, like the path of a ball being thrown, the vertex represents the ball's highest point.
For any quadratic function given by the form \( ax^2 + bx + c \), the formula to find the \( x \)-coordinate of the vertex is \( -\frac{b}{2a} \).
By substituting our values, \( a = -0.8 \) and \( b = 2.4 \), we calculate:

• \( -\frac{2.4}{2 \times (-0.8)} = 1.5 \)

This implies that at a horizontal distance of 1.5 feet, the ball reaches its maximum height.
Once we have the \( x \)-coordinate, we can find the maximum height by evaluating the original function at \( x = 1.5 \). This gives us the height at that point:

• \( f(1.5) = -0.8 \times (1.5)^2 + 2.4 \times 1.5 + 6 = 7.2 \) feet

This maximum height is crucial for understanding projectile motion, as it affects how far and how high the object travels.

Quadratic Formula

Quadratic formulas help find the roots of a quadratic equation, which are the \( x \)-values where the quadratic function equals zero. These values are particularly useful to establish where a projectile lands.
In this problem, the quadratic formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Plugging in our values, where \( a = -0.8 \), \( b = 2.4 \), and \( c = 6 \), the discriminant is positive, ensuring two real roots:

• \( b^2 - 4ac = (2.4)^2 - 4 \times (-0.8) \times 6 = 13.44 \)

Calculating the roots, only the positive root is meaningful for projectile motion, as it measures how far the projectile travels:

• \( x = 4.2 \) feet

This root represents the horizontal distance from the throw's origin to where the ball lands. The quadratic formula is reliable for solving real-world scenarios, offering precise distance calculations even in a complex trajectory.

Projectile Motion

Projectile motion refers to the path of an object that is launched into the air and is influenced only by gravity and its initial velocity. This path typically takes the shape of a parabola.

• The highest point of the parabola is the vertex, providing the maximum height, as calculated in earlier sections.
• The horizontal distance or range is determined by the projectile's initial speed and angle, reflected in the parabola's roots.

In our example, the projectile follows a quadratic trajectory from an initial height of 6 feet. Its path is influenced by both its upward velocity and gravity's downward pull.
The graph of the function provides a visual representation of the ball's journey, tracing from its release point, rising to the peak at 1.5 feet horizontally and 7.2 feet up, and finally descending back to the ground at 4.2 feet from the origin.
Understanding projectile motion through functions helps predict trajectories, critical in numerous fields such as sports, engineering, and physics.