Answer :
To determine the interval of time for which Jerald is less than 104 feet above the ground, we start with the given height equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find the value of [tex]\( t \)[/tex] for which [tex]\( h < 104 \)[/tex]. First, we can set up the equation:
[tex]\[ -16t^2 + 729 = 104 \][/tex]
We solve for [tex]\( t \)[/tex]:
1. Subtract 104 from both sides of the equation:
[tex]\[ -16t^2 + 729 - 104 = 0 \][/tex]
[tex]\[ -16t^2 + 625 = 0 \][/tex]
2. Simplify the equation:
[tex]\[ -16t^2 = -625 \][/tex]
3. Divide both sides by -16:
[tex]\[ t^2 = \frac{625}{16} \][/tex]
4. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \pm \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t = \pm \frac{25}{4} \][/tex]
[tex]\[ t = \pm 6.25 \][/tex]
Thus, we have [tex]\( t = 6.25 \)[/tex] and [tex]\( t = -6.25 \)[/tex].
However, since [tex]\( t \)[/tex] represents time in seconds and cannot be negative, we only consider the positive value: [tex]\( t = 6.25 \)[/tex].
Therefore, the interval of time for which Jerald's height is less than 104 feet above the ground is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]
So, the correct answer is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find the value of [tex]\( t \)[/tex] for which [tex]\( h < 104 \)[/tex]. First, we can set up the equation:
[tex]\[ -16t^2 + 729 = 104 \][/tex]
We solve for [tex]\( t \)[/tex]:
1. Subtract 104 from both sides of the equation:
[tex]\[ -16t^2 + 729 - 104 = 0 \][/tex]
[tex]\[ -16t^2 + 625 = 0 \][/tex]
2. Simplify the equation:
[tex]\[ -16t^2 = -625 \][/tex]
3. Divide both sides by -16:
[tex]\[ t^2 = \frac{625}{16} \][/tex]
4. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \pm \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t = \pm \frac{25}{4} \][/tex]
[tex]\[ t = \pm 6.25 \][/tex]
Thus, we have [tex]\( t = 6.25 \)[/tex] and [tex]\( t = -6.25 \)[/tex].
However, since [tex]\( t \)[/tex] represents time in seconds and cannot be negative, we only consider the positive value: [tex]\( t = 6.25 \)[/tex].
Therefore, the interval of time for which Jerald's height is less than 104 feet above the ground is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]
So, the correct answer is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]