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Answer :
We are given Jerald's height as a function of time:
[tex]$$
h(t) = -16t^2 + 729.
$$[/tex]
We want to find the interval of time for which his height is less than 104 feet. This means we need to solve the inequality:
[tex]$$
-16t^2 + 729 < 104.
$$[/tex]
Step 1. Subtract 104 from both sides
Subtract 104 from both sides to set the inequality to less than 0:
[tex]$$
-16t^2 + 729 - 104 < 0 \quad \Longrightarrow \quad -16t^2 + 625 < 0.
$$[/tex]
Step 2. Multiply by [tex]\(-1\)[/tex] and reverse the inequality
Multiplying both sides by [tex]\(-1\)[/tex] (which reverses the inequality sign):
[tex]$$
16t^2 - 625 > 0.
$$[/tex]
Step 3. Solve for [tex]\(t\)[/tex]
Add 625 to both sides in the transformed inequality:
[tex]$$
16t^2 > 625.
$$[/tex]
Divide both sides by [tex]\(16\)[/tex]:
[tex]$$
t^2 > \frac{625}{16}.
$$[/tex]
Take the square root of both sides:
[tex]$$
t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}.
$$[/tex]
Calculating the square root:
[tex]$$
\sqrt{\frac{625}{16}} = \frac{25}{4} = 6.25.
$$[/tex]
Because time [tex]\(t\)[/tex] must be non-negative, the solution [tex]\(t < -6.25\)[/tex] is not applicable. Thus, we have:
[tex]$$
t > 6.25.
$$[/tex]
Conclusion
Jerald is less than 104 feet above the ground for times [tex]\(t > 6.25\)[/tex] seconds.
[tex]$$
h(t) = -16t^2 + 729.
$$[/tex]
We want to find the interval of time for which his height is less than 104 feet. This means we need to solve the inequality:
[tex]$$
-16t^2 + 729 < 104.
$$[/tex]
Step 1. Subtract 104 from both sides
Subtract 104 from both sides to set the inequality to less than 0:
[tex]$$
-16t^2 + 729 - 104 < 0 \quad \Longrightarrow \quad -16t^2 + 625 < 0.
$$[/tex]
Step 2. Multiply by [tex]\(-1\)[/tex] and reverse the inequality
Multiplying both sides by [tex]\(-1\)[/tex] (which reverses the inequality sign):
[tex]$$
16t^2 - 625 > 0.
$$[/tex]
Step 3. Solve for [tex]\(t\)[/tex]
Add 625 to both sides in the transformed inequality:
[tex]$$
16t^2 > 625.
$$[/tex]
Divide both sides by [tex]\(16\)[/tex]:
[tex]$$
t^2 > \frac{625}{16}.
$$[/tex]
Take the square root of both sides:
[tex]$$
t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}}.
$$[/tex]
Calculating the square root:
[tex]$$
\sqrt{\frac{625}{16}} = \frac{25}{4} = 6.25.
$$[/tex]
Because time [tex]\(t\)[/tex] must be non-negative, the solution [tex]\(t < -6.25\)[/tex] is not applicable. Thus, we have:
[tex]$$
t > 6.25.
$$[/tex]
Conclusion
Jerald is less than 104 feet above the ground for times [tex]\(t > 6.25\)[/tex] seconds.
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