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Answer :
To find the interval of time for which Jerald is less than 104 feet above the ground, we can use the given height equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when Jerald's height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
1. Rearrange the inequality:
First, subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
Simplify:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Move terms to isolate the quadratic expression:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide the entire inequality by -16:
When dividing by a negative number, remember to reverse the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Simplify the right side:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides:
Since we are working with time, we'll consider the positive root for t:
[tex]\[ t > \sqrt{39.0625} \][/tex]
Calculating the square root gives:
[tex]\[ t > 6.25 \][/tex]
So, Jerald is less than 104 feet above the ground for the interval where [tex]\( t > 6.25 \)[/tex]. The answer is [tex]\( t > 6.25 \)[/tex].
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when Jerald's height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
1. Rearrange the inequality:
First, subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
Simplify:
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Move terms to isolate the quadratic expression:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide the entire inequality by -16:
When dividing by a negative number, remember to reverse the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Simplify the right side:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides:
Since we are working with time, we'll consider the positive root for t:
[tex]\[ t > \sqrt{39.0625} \][/tex]
Calculating the square root gives:
[tex]\[ t > 6.25 \][/tex]
So, Jerald is less than 104 feet above the ground for the interval where [tex]\( t > 6.25 \)[/tex]. The answer is [tex]\( t > 6.25 \)[/tex].
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