6.6 Q-6

Question Statement

Two listening posts hear the sound of an enemy gun, with the difference in time being one second. The listening posts are 1400 feet apart. Write the equation of the hyperbola passing through the position of the enemy gun, given that the speed of sound is $1080 , \text{ft/sec}$.

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Background and Explanation

The problem involves determining the locus of points (the hyperbola) where the difference in distances from two fixed points (listening posts) is constant. This is a defining property of a hyperbola.

• The two listening posts represent the foci of the hyperbola.
• The difference in distances corresponds to the time delay multiplied by the speed of sound ($1080 , \text{ft/sec}$).
• The given distance between the listening posts allows us to determine $2c$, the distance between the foci.

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Solution

Step 1: Define key parameters

• The listening posts are 1400 feet apart, so:

$$2c = 1400 \quad \Rightarrow \quad c = 700.$$

• The difference in times is 1 second, and the speed of sound is $1080 , \text{ft/sec}$. Thus:

$$2a = 1080 \quad \Rightarrow \quad a = \frac{1080}{2} = 540.$$

Step 2: Relating $a$, $b$, and $c$

For a hyperbola:

$$c^2 = a^2 + b^2.$$

Substituting the known values of $c$ and $a$:

$$700^2 = 540^2 + b^2.$$

Step 3: Solve for $b^2$

$$490000 = 291600 + b^2,$$ $$b^2 = 490000 - 291600 = 198400.$$

Step 4: Write the equation of the hyperbola

The standard form of the equation is:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.$$

Substituting $a^2 = 540^2 = 291600$ and $b^2 = 198400$:

$$\frac{x^2}{291600} - \frac{y^2}{198400} = 1.$$

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Key Formulas or Methods Used

1. Hyperbolic Property:

$$|PS' - PS| = 2a.$$

2. Speed-Distance Relation:

$$\text{Distance difference} = \text{Speed of sound} \times \text{Time difference}.$$

3. Relation in a Hyperbola:

$$c^2 = a^2 + b^2.$$

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Summary of Steps

1. Determine $c$: The distance between the listening posts is $2c = 1400 , \text{ft}$, so $c = 700 , \text{ft}$.
2. Determine $a$: The difference in time gives $2a = 1080 , \text{ft}$, so $a = 540 , \text{ft}$.
3. Find $b^2$: Use the relation $c^2 = a^2 + b^2$ to compute $b^2 = 198400$.
4. Write the equation: Substitute $a^2$ and $b^2$ into the hyperbolic equation to get:

$$\frac{x^2}{291600} - \frac{y^2}{198400} = 1.$$