5) You and a friend live 4 miles apart. You hear the sound of thunder 18 seconds before your friend hears it. Assuming sound travels at 1100 feet per second, determine where the lightning occur?

At 1100 ft per second for 18 seconds, the sound travels 19,800 ft, or 3.75 miles farther to my friend's house. The set of points that lie 3.75 miles farther from my friend's house than from my house form a hyperbolic curve. This is illustrated by the blue line in the attached graph. (My house is the red dot on the left; my friend's house is the red dot on the right.)

The lightning occurred somewhere on the blue curve.

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If the lightning occurred on the line between our houses, it was 1/8 mile from my house and 3 7/8 mile from my friend's house. (That's close!)

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The formula for the curve in the graph is the distance formula applied to the set of points (x, y). It equates the difference of distance from the two houses to 3.75 miles. If one were to write the equation of the hyperbola in standard form, the equation would look a little different and a restriction would need to be applied so the formula would describe only one branch of the hyperbola.