Answer :
To solve the problem of finding the time interval where Jerald's height is less than 104 feet, we start by using the given equation modeling his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine for which values of [tex]\( t \)[/tex] the height [tex]\( h \)[/tex] is less than 104 feet.
1. Set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
2. Move constants around to isolate the [tex]\( t^2 \)[/tex] term:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16 and remember to reverse the inequality sign because you are dividing by a negative number:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
This calculation shows that Jerald's height is less than 104 feet when the time [tex]\( t \)[/tex] is greater than 6.25 seconds. Therefore, the correct interval is:
[tex]\( t > 6.25 \)[/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine for which values of [tex]\( t \)[/tex] the height [tex]\( h \)[/tex] is less than 104 feet.
1. Set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
2. Move constants around to isolate the [tex]\( t^2 \)[/tex] term:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16 and remember to reverse the inequality sign because you are dividing by a negative number:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
This calculation shows that Jerald's height is less than 104 feet when the time [tex]\( t \)[/tex] is greater than 6.25 seconds. Therefore, the correct interval is:
[tex]\( t > 6.25 \)[/tex]
