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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]

Answer :

To solve this problem, we need to determine when Jerald's height is less than 104 feet above the ground using the equation [tex]\( h = -16t^2 + 729 \)[/tex].

1. Set up the inequality:

We need to find for which values of [tex]\( t \)[/tex], [tex]\( h < 104 \)[/tex].
[tex]\[
-16t^2 + 729 < 104
\][/tex]

2. Subtract 104 from both sides:

[tex]\[
-16t^2 + 729 - 104 < 0
\][/tex]
[tex]\[
-16t^2 + 625 < 0
\][/tex]

3. Rearrange the terms:

[tex]\[
16t^2 > 625
\][/tex]

4. Divide by 16:

[tex]\[
t^2 > \frac{625}{16}
\][/tex]

5. Calculate the square root:

[tex]\[
t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}
\][/tex]

Simplifying the square root:
[tex]\[
t > \frac{25}{4} \quad \text{or} \quad t < -\frac{25}{4}
\][/tex]

6. Calculate [tex]\( \frac{25}{4} \)[/tex]:

[tex]\[
\frac{25}{4} = 6.25
\][/tex]

So, the solutions are:
[tex]\[
t > 6.25 \quad \text{or} \quad t < -6.25
\][/tex]

However, since time [tex]\( t \)[/tex] typically starts from [tex]\( t = 0 \)[/tex] (because negative time isn't logical in this context), the meaningful solution is:
[tex]\[
t > 6.25
\][/tex]

Thus, Jerald is less than 104 feet above the ground for the time interval:
[tex]\[
t > 6.25
\][/tex]

So, the correct answer is [tex]\( t > 6.25 \)[/tex].

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