A piece of rope falls out of a hot air balloon from a height of 5,184 ft. If the
equation for height as a function of time
is h(t) = -16t2 - initial height where t is
time in seconds and h(t) is height in feet,
how many seconds will it take for the
piece of rope to hit the ground?

18 seconds

Step-by-step explanation:

Given the equation for height as a function of time expressed as

h(t) = -16t² - initial height, if initial height is 5,184ft, the expression will become;

h(t) = -16t² - 5,184

t is  time in seconds

h(t) is height in feet

The height of the rope on the ground is zero. In order to calculate the amount of seconds it will take the  piece of rope to hit the ground, we will substitute h(t) = 0 into the modeled equation as shown:

Since the body falls downwards (negative direction), initial height = -5184 ft

h(t) = -16t² - -(5,184)

0 = -16t² + 5,184

0+16t² = 5184

16t² = 5184

16t²/16 = 5184/16

t² = 324

t = √324

t = 18 seconds

Hence, it will take the rope 18seconds to hit the ground.