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Answer :
To solve the problem of determining the interval of time during which Jerald is less than 104 feet above the ground after jumping from a bungee tower, let's analyze the given equation and solve for the interval of interest.
The equation modeling Jerald's height is:
[tex]\[ h = -16t^2 + 729 \][/tex]
We are asked to find when Jerald's height is less than 104 feet. Thus, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this inequality step-by-step:
1. Subtract 104 from both sides of the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Rearrange the terms to isolate [tex]\(-16t^2\)[/tex]:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide every term by -16 to solve for [tex]\(t^2\)[/tex], remembering to flip the inequality sign because we are dividing by a negative number:
[tex]\[ t^2 > \frac{-625}{-16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
4. Take the square root of both sides to solve for [tex]\(t\)[/tex]:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
Based on these steps, Jerald is less than 104 feet above the ground when the time [tex]\(t\)[/tex] is greater than 6.25 seconds. Therefore, the correct interval of time when Jerald's height is below 104 feet is:
[tex]\[ t > 6.25 \][/tex]
The equation modeling Jerald's height is:
[tex]\[ h = -16t^2 + 729 \][/tex]
We are asked to find when Jerald's height is less than 104 feet. Thus, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this inequality step-by-step:
1. Subtract 104 from both sides of the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Rearrange the terms to isolate [tex]\(-16t^2\)[/tex]:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide every term by -16 to solve for [tex]\(t^2\)[/tex], remembering to flip the inequality sign because we are dividing by a negative number:
[tex]\[ t^2 > \frac{-625}{-16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
4. Take the square root of both sides to solve for [tex]\(t\)[/tex]:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
Based on these steps, Jerald is less than 104 feet above the ground when the time [tex]\(t\)[/tex] is greater than 6.25 seconds. Therefore, the correct interval of time when Jerald's height is below 104 feet is:
[tex]\[ t > 6.25 \][/tex]
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