Answer :
To determine the interval of time during which Jerald is less than 104 feet above the ground, we need to solve the inequality given by the height equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when his height is less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Here's how we solve this step-by-step:
1. Rearrange the Inequality:
- Subtract 104 from both sides to isolate the terms involving [tex]\( t \)[/tex]:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
2. Simplify the Expression:
- Simplify the constant terms:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
3. Solve the Quadratic Inequality:
- To find where the parabola represented by the equation [tex]\( -16t^2 + 625 = 0 \)[/tex] crosses the time axis (critical points), we solve:
[tex]\[ -16t^2 + 625 = 0 \][/tex]
4. Calculate the Critical Points:
- Rearrange to solve for [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = \frac{625}{16} \][/tex]
5. Find the Values of [tex]\( t \)[/tex]:
- Take the square root of both sides to find the values of [tex]\( t \)[/tex]:
[tex]\[ t = \pm \sqrt{\frac{625}{16}} \][/tex]
6. Solve for [tex]\( t \)[/tex]:
- Simplifying gives:
[tex]\[ t = \pm 6.25 \][/tex]
7. Determine the Interval:
- The values [tex]\( t = -6.25 \)[/tex] and [tex]\( t = 6.25 \)[/tex] are the points where Jerald's height is exactly 104 feet.
- Since the parabola opens downwards (as indicated by the negative coefficient of [tex]\( t^2 \)[/tex]), Jerald's height is less than 104 feet between these two points.
Therefore, Jerald is less than 104 feet above the ground during the time interval:
[tex]\(-6.25 < t < 6.25\)[/tex]
However, since time cannot be negative in this context, we are only concerned with the part of the interval for positive time. Hence, the interval can be considered as:
[tex]\[ 0 < t < 6.25 \][/tex]
Relating it to the options given:
[tex]\[ \boxed{0 \leq t \leq 6.25} \][/tex]
is the correct interval during which Jerald is below 104 feet above the ground.
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when his height is less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Here's how we solve this step-by-step:
1. Rearrange the Inequality:
- Subtract 104 from both sides to isolate the terms involving [tex]\( t \)[/tex]:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
2. Simplify the Expression:
- Simplify the constant terms:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
3. Solve the Quadratic Inequality:
- To find where the parabola represented by the equation [tex]\( -16t^2 + 625 = 0 \)[/tex] crosses the time axis (critical points), we solve:
[tex]\[ -16t^2 + 625 = 0 \][/tex]
4. Calculate the Critical Points:
- Rearrange to solve for [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = \frac{625}{16} \][/tex]
5. Find the Values of [tex]\( t \)[/tex]:
- Take the square root of both sides to find the values of [tex]\( t \)[/tex]:
[tex]\[ t = \pm \sqrt{\frac{625}{16}} \][/tex]
6. Solve for [tex]\( t \)[/tex]:
- Simplifying gives:
[tex]\[ t = \pm 6.25 \][/tex]
7. Determine the Interval:
- The values [tex]\( t = -6.25 \)[/tex] and [tex]\( t = 6.25 \)[/tex] are the points where Jerald's height is exactly 104 feet.
- Since the parabola opens downwards (as indicated by the negative coefficient of [tex]\( t^2 \)[/tex]), Jerald's height is less than 104 feet between these two points.
Therefore, Jerald is less than 104 feet above the ground during the time interval:
[tex]\(-6.25 < t < 6.25\)[/tex]
However, since time cannot be negative in this context, we are only concerned with the part of the interval for positive time. Hence, the interval can be considered as:
[tex]\[ 0 < t < 6.25 \][/tex]
Relating it to the options given:
[tex]\[ \boxed{0 \leq t \leq 6.25} \][/tex]
is the correct interval during which Jerald is below 104 feet above the ground.
