Jerald jumped from a bungee tower. 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 are given the height equation

[tex]$$
h = -16t^2 + 729,
$$[/tex]

and we want to find the time interval when the height is less than 104 feet, i.e., when

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

A good first step is to find when the height is exactly 104 feet. Set

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

Subtract 104 from both sides to isolate the [tex]$t^2$[/tex] term:

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

Simplify the right side:

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

Divide both sides by [tex]$-16$[/tex] (noting that dividing by a negative number will flip the sign):

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

Taking the square root of both sides gives

[tex]$$
t = \sqrt{\frac{625}{16}} = \frac{25}{4} = 6.25.
$$[/tex]

Since [tex]$t$[/tex] represents time, we consider only the positive solution ([tex]$t \ge 0$[/tex]).

Next, we analyze the inequality. The height equation represents a downward-opening parabola (because the coefficient of [tex]$t^2$[/tex] is negative). This means that for times before [tex]$t = 6.25$[/tex] seconds, Jerald’s height is above 104 feet, and after [tex]$t = 6.25$[/tex] seconds, his height falls below 104 feet.

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

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

The correct interval, considering the domain of time ([tex]$t \ge 0$[/tex]), is

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