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Answer :
To solve the problem of determining when Jerald is less than 104 feet above the ground, given the height equation [tex]\( h = -16t^2 + 729 \)[/tex], we can follow these steps:
1. Set Up the Inequality:
We need to find when Jerald's height is less than 104 feet. So, we set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the Inequality:
Subtract 729 from both sides to isolate the quadratic term:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
3. Solve for [tex]\( t^2 \)[/tex]:
Divide both sides by [tex]\(-16\)[/tex]. Remember, when you divide by a negative number, the inequality sign changes direction:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. Find the Critical Points:
Calculate the square root of [tex]\(\frac{625}{16}\)[/tex] to find the critical points:
[tex]\[
t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}
\][/tex]
Simplifying, we find:
[tex]\[
t > 6.25 \quad \text{or} \quad t < -6.25
\][/tex]
5. Determine the Valid Time Interval:
Since time [tex]\( t \)[/tex] cannot be negative (because negative time doesn't make sense in this context), we ignore [tex]\( t < -6.25 \)[/tex].
Therefore, the interval of time for which Jerald is less than 104 feet above the ground is:
[tex]\[
t > 6.25
\][/tex]
Thus, the correct option for the interval of time is [tex]\( t > 6.25 \)[/tex].
1. Set Up the Inequality:
We need to find when Jerald's height is less than 104 feet. So, we set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the Inequality:
Subtract 729 from both sides to isolate the quadratic term:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
3. Solve for [tex]\( t^2 \)[/tex]:
Divide both sides by [tex]\(-16\)[/tex]. Remember, when you divide by a negative number, the inequality sign changes direction:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. Find the Critical Points:
Calculate the square root of [tex]\(\frac{625}{16}\)[/tex] to find the critical points:
[tex]\[
t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}
\][/tex]
Simplifying, we find:
[tex]\[
t > 6.25 \quad \text{or} \quad t < -6.25
\][/tex]
5. Determine the Valid Time Interval:
Since time [tex]\( t \)[/tex] cannot be negative (because negative time doesn't make sense in this context), we ignore [tex]\( t < -6.25 \)[/tex].
Therefore, the interval of time for which Jerald is less than 104 feet above the ground is:
[tex]\[
t > 6.25
\][/tex]
Thus, the correct option for the interval of time is [tex]\( t > 6.25 \)[/tex].
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