Let's start by examining the given problem. We have a barometer falling from a weather balloon at an initial height of [tex]$14,400$[/tex] feet, and the height as a function of time is given by the equation:
[tex]\[ h(t) = -16t^2 + \text{initial height} \][/tex]
The equation represents the height [tex]\(h(t)\)[/tex] in feet after [tex]\(t\)[/tex] seconds. For the barometer to hit the ground, the height [tex]\(h(t)\)[/tex] must be zero. Setting [tex]\(h(t) = 0\)[/tex] and knowing the initial height is [tex]$14,400$[/tex] feet, we get:
[tex]\[ 0 = -16t^2 + 14400 \][/tex]
Now we need to solve this equation for [tex]\(t\)[/tex]. Let's go through the steps:
1. Move [tex]\(14400\)[/tex] to the other side of the equation:
[tex]\[ -16t^2 = -14400 \][/tex]
2. Divide both sides by [tex]\(-16\)[/tex]:
[tex]\[ t^2 = \frac{14400}{16} \][/tex]
3. Calculate the quotient [tex]\(\frac{14400}{16}\)[/tex]:
[tex]\[ t^2 = 900 \][/tex]
4. Now take the square root of both sides to solve for [tex]\(t\)[/tex]:
[tex]\[ t = \sqrt{900} \][/tex]
Since time cannot be negative, we take the positive square root:
[tex]\[ t = 30 \][/tex]
Therefore, it will take [tex]\(30\)[/tex] seconds for the barometer to hit the ground.
