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Answer :
Sure, I'd be happy to help explain how to solve this problem step-by-step.
We need to determine for which interval of time Jerald's height is less than 104 feet above the ground, given the equation for his height: [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( t \)[/tex] is time in seconds.
Step 1: Set up the inequality.
We want Jerald's height to be less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step 2: Solve the inequality.
1. Rearrange the inequality:
[tex]\[
-16t^2 + 729 - 104 < 0
\][/tex]
[tex]\[
-16t^2 + 625 < 0
\][/tex]
2. Simplify and solve for [tex]\( t^2 \)[/tex]:
[tex]\[
16t^2 > 625
\][/tex]
3. Divide both sides by 16:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. Take the square root of both sides:
[tex]\[
t > \sqrt{\frac{625}{16}}
\][/tex]
[tex]\[
t > 6.25
\][/tex]
Step 3: Consider the context.
Since time [tex]\( t \)[/tex] cannot be negative, the only interval for which Jerald is less than 104 feet above the ground is [tex]\( t > 6.25 \)[/tex].
Conclusion:
Jerald's height will be less than 104 feet after [tex]\( t = 6.25 \)[/tex] seconds. Therefore, the correct interval is [tex]\( t > 6.25 \)[/tex].
We need to determine for which interval of time Jerald's height is less than 104 feet above the ground, given the equation for his height: [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( t \)[/tex] is time in seconds.
Step 1: Set up the inequality.
We want Jerald's height to be less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step 2: Solve the inequality.
1. Rearrange the inequality:
[tex]\[
-16t^2 + 729 - 104 < 0
\][/tex]
[tex]\[
-16t^2 + 625 < 0
\][/tex]
2. Simplify and solve for [tex]\( t^2 \)[/tex]:
[tex]\[
16t^2 > 625
\][/tex]
3. Divide both sides by 16:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. Take the square root of both sides:
[tex]\[
t > \sqrt{\frac{625}{16}}
\][/tex]
[tex]\[
t > 6.25
\][/tex]
Step 3: Consider the context.
Since time [tex]\( t \)[/tex] cannot be negative, the only interval for which Jerald is less than 104 feet above the ground is [tex]\( t > 6.25 \)[/tex].
Conclusion:
Jerald's height will be less than 104 feet after [tex]\( t = 6.25 \)[/tex] seconds. Therefore, the correct interval is [tex]\( t > 6.25 \)[/tex].
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