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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 examine the height equation:
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We're looking for the time [tex]\( t \)[/tex] when the height [tex]\( h(t) \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's solve the inequality:
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:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16 (note that dividing by a negative number reverses the inequality):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Calculate [tex]\(\frac{625}{16}\)[/tex]:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
So, Jerald is less than 104 feet above the ground for times greater than 6.25 seconds. Hence, the correct interval of time is:
[tex]\[ t > 6.25 \][/tex]
Therefore, the answer is [tex]\( t > 6.25 \)[/tex].
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We're looking for the time [tex]\( t \)[/tex] when the height [tex]\( h(t) \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's solve the inequality:
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:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16 (note that dividing by a negative number reverses the inequality):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Calculate [tex]\(\frac{625}{16}\)[/tex]:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
So, Jerald is less than 104 feet above the ground for times greater than 6.25 seconds. Hence, the correct interval of time is:
[tex]\[ t > 6.25 \][/tex]
Therefore, the answer is [tex]\( t > 6.25 \)[/tex].
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