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Answer :
To determine the time interval during which Jerald is less than 104 feet above the ground, we need to analyze the equation for his height:
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We want to find when Jerald's height is less than 104 feet, so we set up the inequality:
[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]
Which simplifies to:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Now, solve the inequality [tex]\(-16t^2 + 625 < 0\)[/tex].
Rearrange it to:
[tex]\[ 16t^2 > 625 \][/tex]
3. Divide both sides by 16 to isolate [tex]\(t^2\)[/tex]:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Calculate the value [tex]\( \frac{625}{16} = \left(\frac{25}{4}\right)^2\)[/tex].
4. Take the square root of both sides to find [tex]\(t\)[/tex]:
[tex]\[ t > \frac{25}{4} \quad \text{or} \quad t < -\frac{25}{4} \][/tex]
The solutions to the inequality are [tex]\(t > 6.25\)[/tex] or [tex]\(t < -6.25\)[/tex].
Since time cannot be negative when Jerald jumps, we only consider the positive values. Thus, the interval where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
This means after 6.25 seconds, Jerald drops below 104 feet above the ground.
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We want to find when Jerald's height is less than 104 feet, so we set up the inequality:
[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]
Which simplifies to:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Now, solve the inequality [tex]\(-16t^2 + 625 < 0\)[/tex].
Rearrange it to:
[tex]\[ 16t^2 > 625 \][/tex]
3. Divide both sides by 16 to isolate [tex]\(t^2\)[/tex]:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Calculate the value [tex]\( \frac{625}{16} = \left(\frac{25}{4}\right)^2\)[/tex].
4. Take the square root of both sides to find [tex]\(t\)[/tex]:
[tex]\[ t > \frac{25}{4} \quad \text{or} \quad t < -\frac{25}{4} \][/tex]
The solutions to the inequality are [tex]\(t > 6.25\)[/tex] or [tex]\(t < -6.25\)[/tex].
Since time cannot be negative when Jerald jumps, we only consider the positive values. Thus, the interval where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
This means after 6.25 seconds, Jerald drops below 104 feet above the ground.
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