Answer :
To find the interval of time during which 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 determine when Jerald's height [tex]\( h \)[/tex] is less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, we solve the equation to find when Jerald's height is exactly 104 feet:
1. Set the equation equal to 104:
[tex]\[ -16t^2 + 729 = 104 \][/tex]
2. Subtract 729 from both sides:
[tex]\[ -16t^2 = 104 - 729 \][/tex]
[tex]\[ -16t^2 = -625 \][/tex]
3. Divide both sides by -16 to solve for [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = \frac{625}{16} \][/tex]
4. Take the square root of both sides to find [tex]\( t \)[/tex]:
[tex]\[ t = \pm \frac{25}{4} \][/tex]
This gives us the critical points [tex]\( t = \frac{25}{4} \)[/tex] and [tex]\( t = -\frac{25}{4} \)[/tex], which are approximately [tex]\( t = 6.25 \)[/tex] and [tex]\( t = -6.25 \)[/tex].
Now, we need to determine when the height is less than 104 feet. The inequality [tex]\( -16t^2 + 729 < 104 \)[/tex] will be true for values of [tex]\( t \)[/tex] that are outside the interval [tex]\(-6.25 < t < 6.25\)[/tex].
Therefore, Jerald is less than 104 feet above the ground for the time intervals:
- [tex]\( t > 6.25 \)[/tex]
- [tex]\( t < -6.25 \)[/tex]
The correct choice from the options provided is:
- [tex]\( t > 6.25 \)[/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when Jerald's height [tex]\( h \)[/tex] is less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, we solve the equation to find when Jerald's height is exactly 104 feet:
1. Set the equation equal to 104:
[tex]\[ -16t^2 + 729 = 104 \][/tex]
2. Subtract 729 from both sides:
[tex]\[ -16t^2 = 104 - 729 \][/tex]
[tex]\[ -16t^2 = -625 \][/tex]
3. Divide both sides by -16 to solve for [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = \frac{625}{16} \][/tex]
4. Take the square root of both sides to find [tex]\( t \)[/tex]:
[tex]\[ t = \pm \frac{25}{4} \][/tex]
This gives us the critical points [tex]\( t = \frac{25}{4} \)[/tex] and [tex]\( t = -\frac{25}{4} \)[/tex], which are approximately [tex]\( t = 6.25 \)[/tex] and [tex]\( t = -6.25 \)[/tex].
Now, we need to determine when the height is less than 104 feet. The inequality [tex]\( -16t^2 + 729 < 104 \)[/tex] will be true for values of [tex]\( t \)[/tex] that are outside the interval [tex]\(-6.25 < t < 6.25\)[/tex].
Therefore, Jerald is less than 104 feet above the ground for the time intervals:
- [tex]\( t > 6.25 \)[/tex]
- [tex]\( t < -6.25 \)[/tex]
The correct choice from the options provided is:
- [tex]\( t > 6.25 \)[/tex]
