Problem 73 - Algebra 1

Question

You are on a research boat in the ocean. You see a penguin jump out of the water. The path followed by the penguin is given by $$ h=-0.05 x^{2}+1.178 x $$ where \(h\) is the height (in feet) the penguin jumps out of the water and \(x\) is the horizontal distance (in feet) traveled by the penguin over the water. How many horizontal feet did the penguin travel over the water before reaching its maximum height?

Step-by-Step Solution

Answer

The penguin travelled approximately 11.78 horizontal feet before reaching its maximum height.

Step 1: Identify the values for a and b in the given equation

In the given quadratic equation \(h=-0.05 x^{2}+1.178 x\), a corresponds to -0.05 and b corresponds to 1.178.

Step 2: Calculate the x-coordinate of the vertex

The x-coordinate of the vertex of a parabola given by the equation: \(ax^2 + bx + c\) is denoted by \(-b/2a\). Thus, for this problem, replace a and b with the values identified in step 1, resulting in \(-1.178 / (2*-0.05)\).

Step 3: Solve for x

After performing the calculation in step 2, we get \(x \approx 11.78\).

Key Concepts

Parabola

A parabola is a U-shaped graph represented by a quadratic equation. It is crucial in understanding projectile paths, like the penguin jumping out of the water. When we talk about a parabola in a mathematical sense, we refer to its equation in the form: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The direction of the parabola is determined by the value of \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards, like in our penguin scenario.

Understanding the basic structure helps when solving quadratic equations, finding maximum or minimum points, and interpreting real-world situations like the penguin’s jump.

Vertex

The vertex is a significant point on the parabola. It represents the peak or the lowest point, depending on the parabola's orientation. For an equation in the standard form \( ax^2 + bx + c \), the vertex formula is crucial. It is provided by the coordinates \((-b/2a, f(-b/2a))\).

• The \(x\)-coordinate of the vertex \((-b/2a)\) shows the horizontal distance for our penguin to reach its maximum height.
• The \(h\)-coordinate provides how high the penguin jumps, calculated by substituting the \(x\)-coordinate into the equation.

In our situation, calculating \(-1.178/(2 * -0.05)\) gives us roughly \(x \approx 11.78\), which is the distance the penguin travels before reaching the highest point.

Maximum Height

Maximum height in the context of a downward-opening parabola represents the peak point, or the highest point the object reaches, before descending. This is particularly useful for applications in physics and engineering.

• For the penguin, the maximum height occurs at the vertex.
• This is an essential concept in determining how far an object can travel before reaching its peak height and starting to descend.

After finding the \(x\)-coordinate of the vertex using \(-b/2a\), substitute this value into the original equation \(h = -0.05x^2 + 1.178x\) to find the maximum height \(h\). So, when \(x \approx 11.78\), replacing it in the equation gives the precise height at that turning point. Understanding this idea aids in predicting trajectory and peak performance in projective motions.