Hot Air Balloon

Question

An object is dropped from a hot-air balloon 1296 feet above the ground. The height of the object is given by

h=-16(t-9)(t+9)

Where the height H is measured in feet, and the time T is measured in seconds. After how many seconds will the object hit the ground????

Answer 1

h = -16(t-9)(t+9)

At t = 0 sec, h = 1296 ft

At t = 9 sec, h = 0 ft

So, after 9 sec the object will hit the ground.

Answer 2

If h - height above the ground then the answer is really t=9 sec. But there is no need to have the function h = 16(t-9(t+9). Just enough to use the number h =1296 as the initial height. In this case the time to reach the ground can be defined by standard formula

                               t = sqrt(2h/g)  where g =32 ft/sec^2 - acceleration due to gravity. Calculations give you the answer t = 9 seconds. I apologize for the confusion in my first response.

                             

Answer 3

The problem states that the height of the object is given by the following equation:

     h = -16(t - 9)(t + 9)

First, let's determine the polynomial equation by expanding it:

     h = -16(t·t + t·9 - 9·t - 9·9)

     h = -16(t² + 9t - 9t - 81)

     h = -16(t² - 81)

     h = (-16)·t² - (-16)·81

     h = -16t² - (-1296)

     h = -16t² + 1296

It is given that the object was dropped from 1296 ft above the ground, which means that when time (t) is equal to 0 seconds that height (h) of the object is equal to 1296 feet. This can be confirmed by plugging in t=0 into the equation above:

     h = -16(0)² + 1296 = -16(0) + 1296 = 0 + 1296 = 1296

The problem asks you to find the time (t) when the object hits the ground, which is to say when the height (h) of the object is equal to 0 feet. So we plug in h=0 into the equation given to solve for t:

     h = -16t² + 1296

     0 = -16t² + 1296

Add '16t²' to both sides of the equation:

     0 + 16t² = -16t² + 1296 + 16t²

     16t² = 1296

Divide both sides of the equation by 16:

     16t²/16 = 1296/16

      t² = 81

Take the square root of both sides of the equation to solve for t:

      √t² = ±√81

        t = ±9

Since time cannot be negative, we ignore the negative solution which leaves us with the solution t = 9 seconds.

Therefore, the object hits the ground (h = 0 ft) after 9 seconds (t = 9 sec).