Answer :
To solve the problem of determining when Jeraid is less than 104 feet above the ground, we start with the height equation given:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find out when this height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's rearrange the inequality:
1. Subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
2. Simplify the left side:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
Next, we need to solve this quadratic inequality:
1. Start by setting the quadratic expression equal to 0 to find the critical points:
[tex]\[ -16t^2 + 625 = 0 \][/tex]
2. Solve for [tex]\( t \)[/tex]:
[tex]\[ -16t^2 = -625 \][/tex]
[tex]\[ t^2 = \frac{625}{16} \][/tex]
[tex]\[ t = \pm \frac{25}{4} \][/tex]
[tex]\[ t = \pm 6.25 \][/tex]
These critical points divide the time axis into intervals. We have:
- [tex]\( t < -6.25 \)[/tex]
- [tex]\( -6.25 < t < 6.25 \)[/tex]
- [tex]\( t > 6.25 \)[/tex]
Next, determine which intervals satisfy the inequality [tex]\(-16t^2 + 625 < 0\)[/tex]:
- For [tex]\( t < -6.25 \)[/tex] and [tex]\( t > 6.25 \)[/tex], substituting points from these intervals into the expression [tex]\(-16t^2 + 625\)[/tex] will result in negative values, indicating that Jeraid's height is less than 104 feet.
Therefore, the intervals where Jeraid is less than 104 feet above the ground are:
- [tex]\( t < -6.25 \)[/tex]
- [tex]\( t > 6.25 \)[/tex]
Since time [tex]\( t \)[/tex] represents seconds elapsed after the jump, we focus on the interval where [tex]\( t > 6.25 \)[/tex].
Thus, the correct interval of time is:
[tex]\[ t > 6.25 \][/tex]
This means Jeraid will be less than 104 feet above the ground after 6.25 seconds.
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find out when this height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's rearrange the inequality:
1. Subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
2. Simplify the left side:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
Next, we need to solve this quadratic inequality:
1. Start by setting the quadratic expression equal to 0 to find the critical points:
[tex]\[ -16t^2 + 625 = 0 \][/tex]
2. Solve for [tex]\( t \)[/tex]:
[tex]\[ -16t^2 = -625 \][/tex]
[tex]\[ t^2 = \frac{625}{16} \][/tex]
[tex]\[ t = \pm \frac{25}{4} \][/tex]
[tex]\[ t = \pm 6.25 \][/tex]
These critical points divide the time axis into intervals. We have:
- [tex]\( t < -6.25 \)[/tex]
- [tex]\( -6.25 < t < 6.25 \)[/tex]
- [tex]\( t > 6.25 \)[/tex]
Next, determine which intervals satisfy the inequality [tex]\(-16t^2 + 625 < 0\)[/tex]:
- For [tex]\( t < -6.25 \)[/tex] and [tex]\( t > 6.25 \)[/tex], substituting points from these intervals into the expression [tex]\(-16t^2 + 625\)[/tex] will result in negative values, indicating that Jeraid's height is less than 104 feet.
Therefore, the intervals where Jeraid is less than 104 feet above the ground are:
- [tex]\( t < -6.25 \)[/tex]
- [tex]\( t > 6.25 \)[/tex]
Since time [tex]\( t \)[/tex] represents seconds elapsed after the jump, we focus on the interval where [tex]\( t > 6.25 \)[/tex].
Thus, the correct interval of time is:
[tex]\[ t > 6.25 \][/tex]
This means Jeraid will be less than 104 feet above the ground after 6.25 seconds.
