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Hyperbola Question:

Two neighbors who live one mile apart hear an explosion while they are talking on the telephone. One neighbor hears the explosion two seconds before the other. If sound travels at 1100 feet
per second, determine the equation of the hyperbola on which the explosion was located.

(i know the answer is
[tex] \frac{ {x}^{2} }{1210000} - \frac{y {}^{2} }{5759600} = 1[/tex]
but i don't understand how you get it)​

Respuesta :

znk

Answer:

[tex]\dfrac{x^{2}}{1210000 } - \dfrac{y^{2}}{5759600} = 1[/tex]

Step-by-step explanation:

A hyperbola is a curve for which the difference of the distances |d₂ - d₁| of any point P from the foci is constant.

The definition leads to the equation

[tex]\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1[/tex]

1. Calculate the value of a²

Assume the neighbours are at the foci A and B.

The explosion is farther from B than from A, so it occurred on the right branch of the hyperbola.

Then AB = 1 mi = 5280 ft

and the distance from the focus to the y-axis is

c = 2640

The vertices are (±a,0).

The speed of sound is 1100 ft/s, so B is 2200 ft further from the explosion.  

The distance from A to the vertex V₁ is c - a. Then

AB = 2200 + 2(c - a) = 2200 -2(2640 - 2a) = 2200 +5280- 2a = 5280

2a = 2200

a = 1100

a² = 1 210 000

2. Calculate the value of b²

[tex]\begin{array}{rcl}a^{2} + b^{2} & = & c^{2}\\1100^{2} + b^{2} & = & 2640^{2}\\b^{2} & = & 2640^{2} - 1100^{2}\\& = & 6969600 - 1210000\\& = & \mathbf{5759600}\\\end{array}[/tex]

3. Write the equation for the hyperbola

[tex]\mathbf{\dfrac{x^{2}}{1210000 } - \dfrac{y^{2}}{5759600}} = \mathbf{1}[/tex]

Ver imagen znk
MrRoyal

The equation of the hyperbola is [tex]\frac{x^2}{1210000} -\frac{y^2}{5759600^2} = 1[/tex]

How to determine the equation of the hyperbola

The equation of an hyperbola is represented as:

[tex]\frac{x^2}{a^2} -\frac{y^2}{b^2} = 1[/tex]

The given parameters are given as:

AB = 1 mile

a = 1100 feet

Express the distance AB as feet

AB = 5280 ft

So, the distance AB from the foci is:

[tex]c = \frac{AB}2[/tex]

This gives

[tex]c = \frac{5280}2[/tex]

[tex]c = 2640[/tex]

The value of b is calculated as:

[tex]a^2 + b^2 = c^2[/tex]

So, we have:

[tex]1100^2 + b^2 = 2640^2[/tex]

This gives

[tex]b^2 = 2640^2 -1100^2[/tex]

Evaluate

[tex]b^2 = 5759600[/tex]

Recall that:

[tex]\frac{x^2}{a^2} -\frac{y^2}{b^2} = 1[/tex]

So, we have:

[tex]\frac{x^2}{1100^2} -\frac{y^2}{5759600^2} = 1[/tex]

[tex]\frac{x^2}{1210000} -\frac{y^2}{5759600^2} = 1[/tex]

Hence, the equation of the hyperbola is [tex]\frac{x^2}{1210000} -\frac{y^2}{5759600^2} = 1[/tex]

Read more about hyperbola at:

https://brainly.com/question/19395889

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