91Ó°ÊÓ

The distance formula is a fundamental concept in geometry. It allows us to calculate the straight-line distance between two points in a plane. This formula is derived from the Pythagorean theorem and is expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where

• \(d\) represents the distance between two points,
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of these points.

In the context of our problem, the distance formula isn't used directly because we focus more on the concept of the hyperbola. However, understanding distances between points, especially along axes, is crucial when defining parts of the hyperbola, such as its foci or the constant difference between distances from any point on the hyperbola to the foci.

Sound velocity, also known as the speed of sound, is the speed at which sound waves travel through a medium. The speed is determined by characteristics of the medium such as density and elasticity. In general,

• The speed of sound in air at room temperature is approximately \(1100 \, \text{feet per second}\) or \(343 \, \text{meters per second}\).
• This speed can change with varying temperature and air pressure.

In this exercise, sound traveling at \(1100 \, \text{feet per second}\) was central because it allowed us to calculate the extra distance sound traveled to reach Samir. Specifically, knowing Anita heard the thunder 0.5 seconds before Samir, we videoed sound traveling an additional \(0.5 \times 1100 = 550 \, \text{feet}\) to reach him. This difference is directly linked to the concept of the hyperbola relating to their positions.

A hyperbola is a type of conic section that arises when a plane intersects both nappes of a double cone. The key property of a hyperbola is that the difference in distances from any point on the hyperbola to the two fixed points called foci is constant. This property leads us to the hyperbola equation.In general, the standard form for a hyperbola centered at the origin with a horizontal transverse axis is:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]Here,

• \(a^2\) is related to the transverse axis length,
• \(b^2\) relates to the conjugate axis,
• The foci are at a distance \(c\) with \(c^2 = a^2 + b^2\).

In the context of the problem, the chosen foci (Anita's and Samir's positions) dictated this equation. The fixed difference of \(550 \, \text{feet}\) corresponds to the standard \(|d_1 - d_2| = (constant)\) form.

The foci of a hyperbola are two specific points on either side of the center of the hyperbola. The unique property of the hyperbola is that for any point on the curve, the absolute difference in distances to the foci is constant. The foci thus play a critical role in defining the exact nature and orientation of the hyperbola.To find the foci of a hyperbola, we use the relationship

• \(c^2 = a^2 + b^2\)

where

• \(c\) is the distance from the center to each focus.

In our exercise, Anita and Samir were assigned the coordinates

• \((-1525, 0)\) and \((1525, 0)\)

respectively, centering them around the coordinate origin. This positioning reflects that they are the foci, spaced \(3050\) feet apart. Understanding the idea of foci is crucial to understanding why the hyperbola's equation is centered around these points.