Jerald jumped from a bungee tower. If the equation that models his height, in feet, is [tex]h = -16t^2 + 729[/tex], where [tex]t[/tex] is the time in seconds, for which interval of time is he less than 104 feet above the ground?

A. [tex]t \ \textgreater \ 6.25[/tex]

B. [tex]-6.25 \ \textless \ t \ \textless \ 6.25[/tex]

C. [tex]t \ \textless \ 6.25[/tex]

D. [tex]0 \leq t \leq 6.25[/tex]

We start with the height function given by

[tex]$$
h(t) = -16t^2 + 729.
$$[/tex]

We need to determine the time interval when Jerald’s height is less than 104 feet. The first step is to find the time when his height is exactly 104 feet. To do this, we set

[tex]$$
-16t^2 + 729 = 104.
$$[/tex]

Subtract 729 from both sides:

[tex]$$
-16t^2 = 104 - 729 = -625.
$$[/tex]

Next, divide both sides by [tex]$-16$[/tex]:

[tex]$$
t^2 = \frac{625}{16}.
$$[/tex]

Taking the square root of both sides (and considering only the positive value since time cannot be negative):

[tex]$$
t = \frac{25}{4} = 6.25 \text{ seconds}.
$$[/tex]

Since the height function describes a downward-opening parabola (because the coefficient of [tex]$ t^2 $[/tex] is negative), Jerald starts at a height of 729 feet and decreases over time. This means that at [tex]$ t = 6.25 $[/tex] seconds his height is exactly 104 feet. For any time greater than [tex]$ 6.25 $[/tex] seconds, his height will be below 104 feet.

Thus, Jerald is less than 104 feet above the ground for

[tex]$$
t > 6.25.
$$[/tex]

Therefore, the correct answer is option 1.