Understanding the Trajectory of a Wolf's Leap: A Mathematical Analysis

Have you ever witnessed an unexpected and stunning leap from a wolf as it bursts out of the dense bushes, leaving a hunter caught off guard? This article delves into the fascinating world of understanding the physics behind such a sudden movement using mathematical equations in the field of physics. We will explore the exact details of a wolf's leap, including how far it leaps and how high it goes, by analyzing the given equation.

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The equation provided that describes the wolf's trajectory is y -x^2 12x - 11, where x and y are measured in feet. To understand the characteristics of this equation, let's break it down and analyze the behavior of the wolf's trajectory.

Understanding the Equations and Graph

The equation y -x^2 12x - 11 is a quadratic equation, and it represents an upside-down parabola. In the context of the wolf's leap, the expression describes a path where the wolf starts at a certain point, jumps upward, and returns to the ground at a different location. The negative coefficient of the x^2 term indicates that the parabola opens downwards, meaning the wolf reaches a peak in its jump and then descends.

To understand the points of significant interest, we need to find the roots of the equation and the vertex of the parabola. The roots represent the points where the wolf touches the ground, and the vertex represents the maximum height of the leap.

Finding the Roots

To find the roots, we solve the equation -x^2 12x - 11 0. By applying the quadratic formula, x [-b ± sqrt(b^2 - 4ac)] / 2a, where a -1, b 12, and c -11, we can determine the values of x.

x [12 ± sqrt((12)^2 - 4(-1)(-11))] / 2(-1)

x [12 ± sqrt(144 - 44)] / -2

x [12 ± sqrt(100)] / -2

x [12 ± 10] / -2

x 1 or x 11

Therefore, the wolf contacts the ground at x 1 and x 11. This means the wolf jumps from 1 to 11 or vice versa, covering a distance of 10 feet.

Max Height of the Leap

To find the maximum height of the leap, we need to locate the vertex of the parabola. The x-coordinate of the vertex of a parabola described by the quadratic equation y ax^2 bx c can be found using the formula x -b / 2a.

x -12 / 2(-1)

x -12 / -2

x 6

Now, we substitute x 6 back into the equation to find the corresponding y-value, which gives us the maximum height of the leap.

y -(6)^2 12(6) - 11

y -36 72 - 11

y 25

Therefore, the wolf reaches a maximum height of 25 feet during its leap.

Conclusion

Through the analysis of the given equation, we can conclude that the wolf leap covers a horizontal distance of 10 feet and reaches a maximum height of 25 feet. This type of mathematical analysis can be applied to many other scenarios in nature and sports, providing us with a deeper understanding of the physics involved in these events.

Key Points to Remember

The roots of the equation represent the points where the wolf contacts the ground. The vertex of the parabola gives the maximum height of the wolf's leap. Mathematics and physics can be used to analyze and understand natural phenomena like a wolf's leap.