Answer :
To determine for which interval of time Jerald is less than 104 feet above the ground, we'll examine the relationship between the height equation and the specified condition.
The height of Jerald is modeled by the equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
where [tex]\( t \)[/tex] is the time in seconds. We want to find when his height [tex]\( h \)[/tex] is less than 104 feet.
Let's set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
1. Rearrange the Inequality: Subtract 104 from both sides to isolate the height expression.
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Reorder to Standard Form: Get the quadratic expression on one side.
[tex]\[ -16t^2 < -625 \][/tex]
3. Solve for [tex]\( t^2 \)[/tex]: Divide by [tex]\(-16\)[/tex] (Note: When you divide by a negative number, the inequality sign flips).
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Take the Square Root: Find the positive square root of both sides since time cannot be negative.
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
5. Calculate the Bound: Simplify [tex]\(\sqrt{\frac{625}{16}}\)[/tex].
[tex]\[ t > \frac{\sqrt{625}}{\sqrt{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
Therefore, Jerald is less than 104 feet above the ground for the interval:
[tex]\[ t > 6.25 \][/tex]
So the correct answer is:
[tex]\[ t > 6.25 \][/tex]
The height of Jerald is modeled by the equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
where [tex]\( t \)[/tex] is the time in seconds. We want to find when his height [tex]\( h \)[/tex] is less than 104 feet.
Let's set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
1. Rearrange the Inequality: Subtract 104 from both sides to isolate the height expression.
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Reorder to Standard Form: Get the quadratic expression on one side.
[tex]\[ -16t^2 < -625 \][/tex]
3. Solve for [tex]\( t^2 \)[/tex]: Divide by [tex]\(-16\)[/tex] (Note: When you divide by a negative number, the inequality sign flips).
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Take the Square Root: Find the positive square root of both sides since time cannot be negative.
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
5. Calculate the Bound: Simplify [tex]\(\sqrt{\frac{625}{16}}\)[/tex].
[tex]\[ t > \frac{\sqrt{625}}{\sqrt{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
Therefore, Jerald is less than 104 feet above the ground for the interval:
[tex]\[ t > 6.25 \][/tex]
So the correct answer is:
[tex]\[ t > 6.25 \][/tex]
