College

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 > 6.25[/tex]
B. [tex]-6.25 < t < 6.25[/tex]
C. [tex]t < 6.25[/tex]
D. [tex]0 \leq t \leq 6.25[/tex]

Answer :

To determine the interval for which Jerald is less than 104 feet above the ground, let's start with the given height equation:

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

We want to find the time [tex]\( t \)[/tex] where Jerald’s height [tex]\( h \)[/tex] is less than 104 feet:

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

First, subtract 104 from both sides to move all terms to one side of the inequality:

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

Next, isolate the quadratic term:

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

Divide both sides by -16 (remember, dividing by a negative number reverses the inequality sign):

[tex]\[ t^2 > \frac{625}{16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]

To solve for [tex]\( t \)[/tex], take the square root of both sides:

[tex]\[ |t| > \sqrt{39.0625} \][/tex]
[tex]\[ |t| > 6.25 \][/tex]

This inequality tells us that [tex]\( t \)[/tex] can be any value greater than 6.25 seconds or less than -6.25 seconds. Since time [tex]\( t \)[/tex] in this context is usually considered non-negative (Jerald cannot go back in time), we discard the negative interval.

Thus, the interval of time for which Jerald is less than 104 feet above the ground is:

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

Therefore, the correct answer is:

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