Answer:
the fireworks display occured at x = 1,266,650 ft
Step-by-step explanation:
We are told that sound travels at 1100 feet per second
We will use the general equation for an hyperbola given by;
(y²/a²) - (x²/b²) = 1
We assume that the longer axis for the hyperbola is on the y axis.
In this problem, the total distance traveled by the sound is 1100 ft per second and we can find the value of "a" as:
2a = 1100
a = 1100/2 = 550
Also, since both people are 5 miles apart, we can find the value of c like this:
2c = 2(5)
So,c = 5 miles
Since the distance of sound was given in ft. Thus;
Lets convert 5 miles to ft.
1 mile = 5280 ft
Thus; 5 miles = 5 x 5280 = 26,400 ft
Now, we can find b² from;
b² = c² - a²
b² = 26400² - 550²
b² = 696960000 - 302500
b² = 696657500
Thus, plugging in the relevant values to obtain;
(y²/302500) - (x²/696657500) = 1
In this case we want to find the value of x when y = 5miles = 26400 ft
solving for x we have:
(26400²/302500) - (x²/696657500) = 1
This leads to;
2304 - (x²/696657500) = 1
Thus;
2304 - 1 = (x²/696657500)
2303 x 696657500 = x²
x² = 1604402222500
x = √1604402222500
x = 1,266,650 ft
