High School

Jerald jumped from a bungee tower. If the equation that models his height, in feet, is [tex]h=-16 t^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 find the interval of time during which Jerald is less than 104 feet above the ground, we need to analyze the given equation that models his height:

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

We're interested in the values of [tex]\( t \)[/tex] for which Jerald's height [tex]\( h \)[/tex] is less than 104 feet:

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

Let's solve this inequality step-by-step:

1. Subtract 104 from both sides:

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

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

2. Rearrange the inequality:

[tex]\[ -16t^2 < -625 \][/tex]

3. Divide both sides by -16. Remember to flip the inequality sign because you're dividing by a negative number:

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

[tex]\[ t^2 > 39.0625 \][/tex]

4. Take the square root of both sides:

[tex]\[ |t| > \sqrt{39.0625} \][/tex]

[tex]\[ |t| > 6.25 \][/tex]

This tells us that the absolute value of [tex]\( t \)[/tex] must be greater than 6.25. In other words, [tex]\( t \)[/tex] must be either greater than 6.25 or less than -6.25:

[tex]\[ t < -6.25 \quad \text{or} \quad t > 6.25 \][/tex]

Given these conditions, Jerald is less than 104 feet above the ground for the following interval of time:

[tex]\((-∞, -6.25) \cup (6.25, ∞)\)[/tex]

So, among the options given, the time interval for which Jerald is less than 104 feet above the ground is:

[tex]\(t > 6.25\)[/tex]