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Answer :
To solve this problem, we need to determine in which interval of time Jerald's height while bungee jumping is less than 104 feet above the ground. The height model given is:
[tex]\[ h = -16t^2 + 729 \][/tex]
where [tex]\( h \)[/tex] is the height in feet and [tex]\( t \)[/tex] is the time in seconds. We are tasked with finding when this height is less than 104 feet.
1. Set up the inequality: We want to find when:
[tex]\[-16t^2 + 729 < 104\][/tex]
2. Isolate the quadratic term: Subtract 104 from both sides to set up the inequality:
[tex]\[-16t^2 + 729 - 104 < 0\][/tex]
[tex]\[-16t^2 + 625 < 0\][/tex]
3. Move the constant term to the other side:
[tex]\[-16t^2 < -625\][/tex]
4. Divide through by -16: When dividing an inequality by a negative number, remember to flip the inequality sign:
[tex]\[t^2 > \frac{625}{16}\][/tex]
5. Find the critical points: Calculate the square roots to find the values of [tex]\( t \)[/tex]:
[tex]\[t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}\][/tex]
[tex]\[\sqrt{\frac{625}{16}} = \frac{\sqrt{625}}{\sqrt{16}} = \frac{25}{4} = 6.25\][/tex]
So, [tex]\( t > 6.25 \)[/tex] or [tex]\( t < -6.25 \)[/tex]
However, since time [tex]\( t \)[/tex] cannot be negative in this context (Jerald cannot be jumping before time [tex]\( t=0 \)[/tex]), we focus only on the positive interval.
6. Determine the valid interval for [tex]\( t \)[/tex]:
- The solution for [tex]\( t \)[/tex] being less than 104 feet is valid when Jerald is actually in the air and has not reached the ground below his starting height. Since [tex]\( t^2 > 6.25^2 \)[/tex] results in Jerald being less than 104 feet, the time is:
[tex]\[0 \leq t \leq 6.25\][/tex]
Therefore, the interval for which Jerald is less than 104 feet above the ground is [tex]\(0 \leq t \leq 6.25\)[/tex], which matches the multiple-choice option [tex]\(0 \leq t \leq 6.25\)[/tex].
[tex]\[ h = -16t^2 + 729 \][/tex]
where [tex]\( h \)[/tex] is the height in feet and [tex]\( t \)[/tex] is the time in seconds. We are tasked with finding when this height is less than 104 feet.
1. Set up the inequality: We want to find when:
[tex]\[-16t^2 + 729 < 104\][/tex]
2. Isolate the quadratic term: Subtract 104 from both sides to set up the inequality:
[tex]\[-16t^2 + 729 - 104 < 0\][/tex]
[tex]\[-16t^2 + 625 < 0\][/tex]
3. Move the constant term to the other side:
[tex]\[-16t^2 < -625\][/tex]
4. Divide through by -16: When dividing an inequality by a negative number, remember to flip the inequality sign:
[tex]\[t^2 > \frac{625}{16}\][/tex]
5. Find the critical points: Calculate the square roots to find the values of [tex]\( t \)[/tex]:
[tex]\[t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}\][/tex]
[tex]\[\sqrt{\frac{625}{16}} = \frac{\sqrt{625}}{\sqrt{16}} = \frac{25}{4} = 6.25\][/tex]
So, [tex]\( t > 6.25 \)[/tex] or [tex]\( t < -6.25 \)[/tex]
However, since time [tex]\( t \)[/tex] cannot be negative in this context (Jerald cannot be jumping before time [tex]\( t=0 \)[/tex]), we focus only on the positive interval.
6. Determine the valid interval for [tex]\( t \)[/tex]:
- The solution for [tex]\( t \)[/tex] being less than 104 feet is valid when Jerald is actually in the air and has not reached the ground below his starting height. Since [tex]\( t^2 > 6.25^2 \)[/tex] results in Jerald being less than 104 feet, the time is:
[tex]\[0 \leq t \leq 6.25\][/tex]
Therefore, the interval for which Jerald is less than 104 feet above the ground is [tex]\(0 \leq t \leq 6.25\)[/tex], which matches the multiple-choice option [tex]\(0 \leq t \leq 6.25\)[/tex].
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