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].
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].