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Answer :
To determine for which interval of time Jerald is less than 104 feet above the ground, we’ll analyze the height equation:
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We need to find when his height [tex]\( h(t) \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
### Solving the Inequality:
1. Isolate the [tex]\( t^2 \)[/tex] term:
Subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
2. Solve for [tex]\( t^2 \)[/tex]:
Divide both sides by -16. Remember, dividing by a negative number flips the inequality sign:
[tex]\[
t^2 > \frac{-625}{-16}
\][/tex]
Simplify the division:
[tex]\[
t^2 > 39.0625
\][/tex]
3. Find [tex]\( t \)[/tex]:
Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
t > \sqrt{39.0625}
\][/tex]
The square root of 39.0625 is approximately 6.25, so:
[tex]\[
t > 6.25
\][/tex]
### Conclusion:
The interval for which Jerald is less than 104 feet above the ground is when [tex]\( t > 6.25 \)[/tex]. Therefore, the correct interval is:
[tex]\[ t > 6.25 \][/tex]
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We need to find when his height [tex]\( h(t) \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
### Solving the Inequality:
1. Isolate the [tex]\( t^2 \)[/tex] term:
Subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
2. Solve for [tex]\( t^2 \)[/tex]:
Divide both sides by -16. Remember, dividing by a negative number flips the inequality sign:
[tex]\[
t^2 > \frac{-625}{-16}
\][/tex]
Simplify the division:
[tex]\[
t^2 > 39.0625
\][/tex]
3. Find [tex]\( t \)[/tex]:
Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
t > \sqrt{39.0625}
\][/tex]
The square root of 39.0625 is approximately 6.25, so:
[tex]\[
t > 6.25
\][/tex]
### Conclusion:
The interval for which Jerald is less than 104 feet above the ground is when [tex]\( t > 6.25 \)[/tex]. Therefore, the correct interval is:
[tex]\[ t > 6.25 \][/tex]
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