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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]

Answer :

Sure, let's solve this problem step-by-step to find out the time interval during which Jerald is less than 104 feet above the ground.

Jerald's height, [tex]\( h \)[/tex], is given by the equation:

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

We want to find the interval of time during which his height is less than 104 feet. So we need to solve the inequality:

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

Step 1: Subtract 104 from both sides

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

Simplify the equation:

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

Step 2: Move 625 to the right side

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

Step 3: Divide both sides by -16

Since we're dividing by a negative number, we need to flip the inequality sign:

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

Step 4: Solve for [tex]\( t \)[/tex]

Calculate the square root of both sides:

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

which simplifies to:

[tex]\[ t > \frac{25}{4} \][/tex]

So we find:

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

This means Jerald's height is less than 104 feet after [tex]\( t > 6.25 \)[/tex] seconds.

Therefore, the correct answer is:

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

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