Answer :
To determine for which interval of time Jerald is less than 104 feet above the ground, we start with the given equation for his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find when Jerald's height, [tex]\( h \)[/tex], is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step-by-step solution:
1. Rearrange the inequality: Subtract 104 from both sides to isolate the terms with [tex]\( t \)[/tex]:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
2. Divide both sides by -16: Remember that dividing by a negative number flips the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
3. Calculate the square root of both sides: Solving for [tex]\( t \)[/tex] requires taking the square root:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{\sqrt{625}}{\sqrt{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
The solution [tex]\( t > 6.25 \)[/tex] implies that Jerald's height is less than 104 feet after 6.25 seconds. Since [tex]\( t \)[/tex] in the context of time must be non-negative and we assumed he started from the height representing the top of the jump, the correct interval where Jerald's height is less than 104 feet is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]
Therefore, Jerald is less than 104 feet above the ground in the time interval [tex]\( 0 \leq t \leq 6.25 \)[/tex].
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find when Jerald's height, [tex]\( h \)[/tex], is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step-by-step solution:
1. Rearrange the inequality: Subtract 104 from both sides to isolate the terms with [tex]\( t \)[/tex]:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
2. Divide both sides by -16: Remember that dividing by a negative number flips the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
3. Calculate the square root of both sides: Solving for [tex]\( t \)[/tex] requires taking the square root:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{\sqrt{625}}{\sqrt{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
The solution [tex]\( t > 6.25 \)[/tex] implies that Jerald's height is less than 104 feet after 6.25 seconds. Since [tex]\( t \)[/tex] in the context of time must be non-negative and we assumed he started from the height representing the top of the jump, the correct interval where Jerald's height is less than 104 feet is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]
Therefore, Jerald is less than 104 feet above the ground in the time interval [tex]\( 0 \leq t \leq 6.25 \)[/tex].
