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Answer :
To find the interval of time during which Jerald is less than 104 feet above the ground, we need to analyze the given equation that models his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We're interested in the values of [tex]\( t \)[/tex] for which Jerald's height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this inequality step-by-step:
1. Subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Rearrange the inequality:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16. Remember to flip the inequality sign because you're 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:
[tex]\[ |t| > \sqrt{39.0625} \][/tex]
[tex]\[ |t| > 6.25 \][/tex]
This tells us that the absolute value of [tex]\( t \)[/tex] must be greater than 6.25. In other words, [tex]\( t \)[/tex] must be either greater than 6.25 or less than -6.25:
[tex]\[ t < -6.25 \quad \text{or} \quad t > 6.25 \][/tex]
Given these conditions, Jerald is less than 104 feet above the ground for the following interval of time:
[tex]\((-∞, -6.25) \cup (6.25, ∞)\)[/tex]
So, among the options given, the time interval for which Jerald is less than 104 feet above the ground is:
[tex]\(t > 6.25\)[/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We're interested in the values of [tex]\( t \)[/tex] for which Jerald's height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this inequality step-by-step:
1. Subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Rearrange the inequality:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16. Remember to flip the inequality sign because you're 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:
[tex]\[ |t| > \sqrt{39.0625} \][/tex]
[tex]\[ |t| > 6.25 \][/tex]
This tells us that the absolute value of [tex]\( t \)[/tex] must be greater than 6.25. In other words, [tex]\( t \)[/tex] must be either greater than 6.25 or less than -6.25:
[tex]\[ t < -6.25 \quad \text{or} \quad t > 6.25 \][/tex]
Given these conditions, Jerald is less than 104 feet above the ground for the following interval of time:
[tex]\((-∞, -6.25) \cup (6.25, ∞)\)[/tex]
So, among the options given, the time interval for which Jerald is less than 104 feet above the ground is:
[tex]\(t > 6.25\)[/tex]
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