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Answer :
To solve the problem of finding when Jerald is less than 104 feet above the ground, we start with the height equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when this height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step 1: Subtract 729 from both sides to set up the inequality:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Step 2: Since we're dividing by a negative number, the inequality sign flips:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Step 3: Find the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{25}{4} = 6.25 \][/tex]
However, because the inequality is [tex]\( t^2 > 625/16 \)[/tex], it implies two conditions:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Since time [tex]\( t \)[/tex] must be positive, we consider:
[tex]\[ t > 6.25 \][/tex]
Thus, the correct interval where Jerald's height is less than 104 feet is:
[tex]\( t > 6.25 \)[/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when this height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step 1: Subtract 729 from both sides to set up the inequality:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Step 2: Since we're dividing by a negative number, the inequality sign flips:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Step 3: Find the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{25}{4} = 6.25 \][/tex]
However, because the inequality is [tex]\( t^2 > 625/16 \)[/tex], it implies two conditions:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Since time [tex]\( t \)[/tex] must be positive, we consider:
[tex]\[ t > 6.25 \][/tex]
Thus, the correct interval where Jerald's height is less than 104 feet is:
[tex]\( t > 6.25 \)[/tex]
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