Answer :
To solve the problem of determining when Jerald is less than 104 feet above the ground given the equation [tex]\( h = -16t^2 + 729 \)[/tex], we start by setting up the inequality based on the height:
1. Set up the inequality:
We want to find when [tex]\( h < 104 \)[/tex]. So, we set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the inequality:
Subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
3. Divide both sides by -16:
Remember, dividing by a negative number will flip the inequality sign:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
Take the square root of both sides. We consider the positive root, as time cannot be negative:
[tex]\[
t > \sqrt{\frac{625}{16}}
\][/tex]
Calculate [tex]\( \sqrt{\frac{625}{16}} \)[/tex]:
[tex]\[
t > 6.25
\][/tex]
5. Determine the interval:
As we calculated, Jerald's height is less than 104 feet when [tex]\( t > 6.25 \)[/tex].
From this analysis, the interval during which Jerald is less than 104 feet above the ground is [tex]\( t > 6.25 \)[/tex].
1. Set up the inequality:
We want to find when [tex]\( h < 104 \)[/tex]. So, we set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the inequality:
Subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
3. Divide both sides by -16:
Remember, dividing by a negative number will flip the inequality sign:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
Take the square root of both sides. We consider the positive root, as time cannot be negative:
[tex]\[
t > \sqrt{\frac{625}{16}}
\][/tex]
Calculate [tex]\( \sqrt{\frac{625}{16}} \)[/tex]:
[tex]\[
t > 6.25
\][/tex]
5. Determine the interval:
As we calculated, Jerald's height is less than 104 feet when [tex]\( t > 6.25 \)[/tex].
From this analysis, the interval during which Jerald is less than 104 feet above the ground is [tex]\( t > 6.25 \)[/tex].