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Answer :
To solve this problem, we need to find the time interval when Jerald's height is less than 104 feet above the ground. The equation modeling Jerald's height is:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find the values of [tex]\( t \)[/tex] for which [tex]\( h < 104 \)[/tex].
### Step-by-Step Solution
1. Set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
2. Subtract 104 from both sides to simplify:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
3. Rearrange to compare with zero:
[tex]\[ 16t^2 > 625 \][/tex]
4. Solve for [tex]\( t^2 \)[/tex]:
Divide both sides by 16 to isolate [tex]\( t^2 \)[/tex].
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Calculate [tex]\( \frac{625}{16} \)[/tex]:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides:
Since [tex]\( t^2 > 39.0625 \)[/tex], we take the square root to find [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
6. Consider the feasible values of [tex]\( t \)[/tex]:
While the mathematical solution suggests [tex]\( t < -\sqrt{39.0625} \)[/tex] as well, in this context (representing time), [tex]\( t \)[/tex] cannot be negative.
7. Conclusion:
Hence, the interval of time for which Jerald's height is less than 104 feet is [tex]\( t > 6.25 \)[/tex].
Therefore, the correct interval is:
[tex]\[ t > 6.25 \][/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to find the values of [tex]\( t \)[/tex] for which [tex]\( h < 104 \)[/tex].
### Step-by-Step Solution
1. Set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
2. Subtract 104 from both sides to simplify:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
3. Rearrange to compare with zero:
[tex]\[ 16t^2 > 625 \][/tex]
4. Solve for [tex]\( t^2 \)[/tex]:
Divide both sides by 16 to isolate [tex]\( t^2 \)[/tex].
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Calculate [tex]\( \frac{625}{16} \)[/tex]:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides:
Since [tex]\( t^2 > 39.0625 \)[/tex], we take the square root to find [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
6. Consider the feasible values of [tex]\( t \)[/tex]:
While the mathematical solution suggests [tex]\( t < -\sqrt{39.0625} \)[/tex] as well, in this context (representing time), [tex]\( t \)[/tex] cannot be negative.
7. Conclusion:
Hence, the interval of time for which Jerald's height is less than 104 feet is [tex]\( t > 6.25 \)[/tex].
Therefore, the correct interval is:
[tex]\[ t > 6.25 \][/tex]
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