Problem 70 - College Algebra

Question

For the object from the previous exercise, assume the path followed is given by \(y=-0.5 x^{2}+80 x\). Determine how far along the horizontal the object traveled to reach maximum height.

Step-by-Step Solution

Answer

The object traveled 80 units horizontally to reach maximum height.

Step 1: Understand the Problem

We need to find the value of \(x\) where the object reaches its maximum height. This means finding the vertex of the parabola given by \(y = -0.5x^{2} + 80x\).

Step 2: Identify the Formula for Vertex

For a quadratic equation in the form \(y = ax^2 + bx + c\), the \(x\)-coordinate of the vertex, which gives the maximum height, is found using the formula \(x = \frac{-b}{2a}\).

Step 3: Substitute the Coefficients

In our equation \(y = -0.5x^2 + 80x\), the coefficients are \(a = -0.5\) and \(b = 80\). Substitute these into the vertex formula: \[x = \frac{-80}{2(-0.5)}\]

Step 4: Solve for x

Calculate the value of \(x\) using the substituted coefficients: \[x = \frac{-80}{-1} = 80\] The object travels 80 units along the horizontal axis to reach its maximum height.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two. This means the highest power of the variable, typically represented as "x", is squared. A standard quadratic equation looks like this:
\[ y = ax^2 + bx + c \] where:

• "a" is the coefficient of the squared term.
• "b" is the coefficient of the linear term.
• "c" is the constant term.

Quadratic equations can model various real-life scenarios, like predicting the trajectory of an object. Solving these equations helps in finding points where the curve meets the x-axis (roots), the vertex, and other key features. Understanding the shape and orientation based on the coefficients is crucial. If "a" is positive, the parabola opens upwards; if negative, it opens downwards, which affects the maximum or minimum values of the function.

Parabola

A parabola is the graph of a quadratic equation and is shaped like a U or an upside-down U, depending on the coefficient "a" of the equation. The general equation for a quadratic function, which produces a parabola, is:
\[ y = ax^2 + bx + c \] The parabola can open upwards or downwards:

• If "a" is positive, the parabola opens upwards, indicating a minimum point.
• If "a" is negative, the parabola opens downwards, suggesting a maximum point.

The vertex of the parabola is its peak or trough, representing either the highest or lowest point. This feature is essential in understanding the parabolic motion in contexts such as projectile motion, where determining the vertex helps find maximum height or distance. Parabolas are also symmetric, with the axis of symmetry passing through the vertex.

Vertex Formula

The vertex formula is a pivotal tool in calculus for quadratic functions. It helps determine the vertex of a parabola, which is especially useful when analyzing curved motion, like projectile paths. The vertex provides either the maximum or minimum value of the quadratic function.
For a quadratic equation given by
\[ y = ax^2 + bx + c \]
the x-component of the vertex is determined by:
\[ x = \frac{-b}{2a} \]

• "b" is the coefficient of the linear term.
• "a" is the coefficient of the squared term.

By using this formula, we can find where the maximum or minimum y-value occurs, thus pinpointing the vertex location on the graph. For example, in the context provided, substituting "a = -0.5" and "b = 80" into the formula gives the maximum height point of the parabolic path.

Maximum Height

The maximum height in the context of a parabolic motion is the highest point reached by an object, usually under the influence of gravity if it's a physical object. This height is found at the vertex of the parabola when the parabola opens downwards.
Using the vertex formula for the equation
\[ y = -0.5x^2 + 80x \], we find:

• The x-coordinate (horizontal distance) at maximum height is \( x = \frac{-80}{-1} = 80 \).

This means the object has traveled 80 units horizontally to reach its peak. Calculating the y-coordinate by substituting \( x = 80 \) back into the equation will give the actual maximum height reached by the object. Understanding this concept is crucial for predicting the behavior of projectiles and optimizing trajectories.