Question

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?

Answer 1

To determine the location of the lightning, we calculate the time it took for the sound to travel the 4 miles between you and your friend. Using the speed of sound, we find that the time is 17.87 seconds.

Step-by-step explanation:

To determine the location of the lightning, we first need to calculate the time it took for the sound to travel the 4 miles between you and your friend. Sound travels at a speed of 1100 feet per second. Since the distance is 4 miles, we need to convert it to feet by multiplying it by 5280 (1 mile = 5280 feet). The distance between you and your friend is then 4 miles * 5280 feet/mile = 21120 feet.

Next, we calculate the time it took for the sound to travel this distance. You heard the sound 18 seconds before your friend did, so we subtract 18 seconds from the total time. The time is then 18 seconds.

Finally, we divide the distance by the time to find the speed of sound. Speed equals distance divided by time, so the speed of sound is 21120 feet / 18 seconds = 1173.33 feet per second.

Since the speed of sound is 1173.33 feet per second, we can calculate the time it took for the sound to travel 4 miles using this speed. The time is distance divided by the speed, so the time is 4 miles * 5280 feet/mile / 1173.33 feet per second = 17.87 seconds.

Answer 2

see attached

Step-by-step explanation:

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.

___

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!)

_____

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.