Problem 69 - College Algebra

Question

An object is projected so as to follow a parabolic path given by \(y=-x^{2}+96 x,\) where \(x\) is the horizontal distance traveled in feet and \(y\) is the height. Determine the maximum height the object reaches.

Step-by-Step Solution

Answer

The maximum height is 2304 feet.

Step 1: Understand the Parabolic Equation

The given equation is a quadratic function of the form \(y = -x^2 + 96x\). This represents a downward-opening parabola, as the coefficient of \(x^2\) is negative. Our goal is to determine the maximum height, which is the vertex of this parabola.

Step 2: Find the Vertex Formula

The vertex of a parabola given by \(y = ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\). In this equation, \(a = -1\) and \(b = 96\).

Step 3: Calculate x-vertex

Substitute \(a = -1\) and \(b = 96\) into the vertex formula: \[x = -\frac{96}{2(-1)} = 48.\] This means the maximum height occurs when \(x = 48\).

Step 4: Find Maximum Height (y-vertex)

Substitute \(x = 48\) back into the original equation to find \(y\): \[y = -(48)^2 + 96(48) = -2304 + 4608 = 2304.\] Thus, the maximum height is 2304 feet.

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two and can be generally expressed as \(y = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, with \(a\) being nonzero. This form of function has a characteristic "U" shape called a parabola.
When \(a > 0\), the parabola opens upwards, resembling a smiling face, while if \(a < 0\), it opens downwards, looking like a frown. Quadratic functions are pivotal in many real-world problems, from physics and engineering to economics, where they describe phenomena like projectile motion, area optimization, and profit maximization.
In the given exercise, the specific quadratic function is \(y = -x^2 + 96x\). Here, \(a = -1\), meaning the parabola opens downward, and we are interested in finding the peak point or maximum height, which occurs at the vertex of the parabola.

Vertex of a Parabola

The vertex of a parabola is a crucial point that represents the maximum or minimum value of a quadratic function. For a parabola expressed as \(y = ax^2 + bx + c\), the formula to find the x-coordinate of the vertex is \(x = -\frac{b}{2a}\).
This formula derives from taking the derivative of the quadratic equation and setting it to zero in order to find where the slope is zero, which is where the maximum or minimum occurs.
In our exercise, the equation is \(y = -x^2 + 96x\), making \(a = -1\) and \(b = 96\). Placing these into the vertex formula gives us \(x = -\frac{96}{2(-1)} = 48\). Thus, the x-coordinate for the vertex and hence, where the maximum height occurs, is 48 feet along the horizontal axis. Once you find this x-coordinate, substitute back into the original quadratic equation to find the corresponding y-coordinate, representing the maximum height.

Parabolic Equation

A parabolic equation often models situations where there is some kind of symmetry in the system, such as projectile motions or suspension bridges. In mathematics, the general form \(y = ax^2 + bx + c\) is called a parabolic equation. The values of \(a\), \(b\), and \(c\) determine the specific characteristics of the parabola, like width, direction, and position.
In the exercise, the parabolic equation \(y = -x^2 + 96x\) is given.

• The negative coefficient \(-1\) in front of \(x^2\) means the parabola opens downward, indicating that it will have a maximum point, not a minimum.
• To find this maximum point, the vertex formula is used to find the x-coordinate, which is then substituted back into the equation to find the y-coordinate.

The y-coordinate of the vertex represents the maximum height of the object's path, which is pivotal for problems involving projectile motion, like figuring out how high a ball or rocket travels.