Answer :
To solve the problem of finding for which interval of time Jerald is less than 104 feet above the ground, we are given the equation for Jerald's height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find the time interval where his height is less than 104 feet. This translates to solving the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this step by step:
1. Set up the inequality:
[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 equation:
[tex]\[ 625 - 16t^2 < 0 \][/tex]
[tex]\[ 16t^2 > 625 \][/tex]
4. Divide both sides by 16:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
5. Simplify the right side:
[tex]\[ t^2 > 39.0625 \][/tex]
6. Take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \quad \text{or} \quad t < -\sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Jerald's height is less than 104 feet in the time intervals where [tex]\( t \)[/tex] is greater than 6.25 seconds or less than -6.25 seconds.
Since time [tex]\( t \)[/tex] cannot be negative because it represents time elapsed from jumping, we only consider positive values of [tex]\( t \)[/tex].
Therefore, the interval of time where Jerald is less than 104 feet above the ground is when:
[tex]\[ t > 6.25 \][/tex]
This corresponds to the option [tex]\( t > 6.25 \)[/tex].
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find the time interval where his height is less than 104 feet. This translates to solving the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this step by step:
1. Set up the inequality:
[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 equation:
[tex]\[ 625 - 16t^2 < 0 \][/tex]
[tex]\[ 16t^2 > 625 \][/tex]
4. Divide both sides by 16:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
5. Simplify the right side:
[tex]\[ t^2 > 39.0625 \][/tex]
6. Take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \quad \text{or} \quad t < -\sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Jerald's height is less than 104 feet in the time intervals where [tex]\( t \)[/tex] is greater than 6.25 seconds or less than -6.25 seconds.
Since time [tex]\( t \)[/tex] cannot be negative because it represents time elapsed from jumping, we only consider positive values of [tex]\( t \)[/tex].
Therefore, the interval of time where Jerald is less than 104 feet above the ground is when:
[tex]\[ t > 6.25 \][/tex]
This corresponds to the option [tex]\( t > 6.25 \)[/tex].
