Problem 59 - Algebra and Trigonometry

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 7.3 feet and this happens at a distance of 1.5 feet from where it was thrown. b. The ball travels horizontally for a distance of 5 feet before hitting the ground. c. For the graph, sketch a downward opening parabola with vertex at (1.5, 7.3) and intercepts at (0,6) and (5,0).

Step 1: Identify The Maximum Height

The maximum height is obtained at the vertex of the parabola. Since the parabola opens downwards (because the leading coefficient is negative), the vertex is the maximum point. The x-coordinate of the vertex, \(h\), is given by the formula \(-b/(2a)\), where \(a\) and \(b\) are the coefficients from the quadratic equation. Substituting values, we get: \[ h = -2.4 / (2*(-0.8)) = 1.5 \] feet. Substitute \(h\) into the equation to find the maximum height, \(k = f(h)\): \[ k = -0.8*(1.5)^2 + 2.4*1.5 + 6 = 7.3 \] feet.

Step 2: Calculate the Projectile's Horizontal Distance

To find when the ball hits the ground, we have to solve the equation \(f(x) = 0\). This will give us the roots of the quadratic equation, which represent the points where the ball hits the ground. Setting up the equation: \[ 0 = -0.8 x^{2} + 2.4 x + 6 \] and solving it using the quadratic formula we find that: \(x = -1.5\) and \(x = 5\). Since a negative distance doesn't make sense in this context, we disregard the negative root, so the horizontal distance before hitting the ground is 5 feet.

Step 3: Graph the Parabola

The graphing step is visual and subjective in nature, and might not be clearly explainable in text form. However, we can give general instructions. The graph should resemble a downward-opening U shape (since the coefficient of \(x^2\) is negative). The vertex of the parabola is at the point (1.5, 7.3), and the parabola intersects the x-axis at points (0, 6) and (5, 0).