Answer :
To find the interval during which Jerald's height is less than 104 feet, we need to solve the inequality given by the equation for his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to determine when this height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this inequality step-by-step:
1. Subtract 104 from both sides:
[tex]\[
-16t^2 + 729 - 104 < 0
\][/tex]
This simplifies to:
[tex]\[
-16t^2 + 625 < 0
\][/tex]
2. Rearrange the terms:
[tex]\[
625 - 16t^2 < 0
\][/tex]
3. Multiply everything by -1 (this reverses the inequality):
[tex]\[
16t^2 - 625 > 0
\][/tex]
4. Factor the quadratic expression:
[tex]\[
(4t - 25)(4t + 25) > 0
\][/tex]
5. Find the critical points by solving [tex]\(4t - 25 = 0\)[/tex] and [tex]\(4t + 25 = 0\)[/tex]:
[tex]\[
4t - 25 = 0 \Rightarrow t = \frac{25}{4} = 6.25
\][/tex]
[tex]\[
4t + 25 = 0 \Rightarrow t = -\frac{25}{4} = -6.25
\][/tex]
6. Determine the intervals to test the inequality:
The critical points divide the number line into three intervals:
[tex]\[
(-\infty, -6.25), \quad (-6.25, 6.25), \quad (6.25, \infty)
\][/tex]
7. Test a point from each interval to see where the inequality holds:
- Interval [tex]\((-6.25, 6.25)\)[/tex]:
Choose [tex]\(t = 0\)[/tex]:
[tex]\[
16(0)^2 - 625 = -625 \Rightarrow \text{False}
\][/tex]
- Interval [tex]\((6.25, \infty)\)[/tex]:
Choose [tex]\(t = 7\)[/tex]:
[tex]\[
16(7)^2 - 625 = 784 - 625 = 159 > 0 \Rightarrow \text{True}
\][/tex]
- Interval [tex]\((-\infty, -6.25)\)[/tex]:
Choose [tex]\(t = -7\)[/tex]:
[tex]\[
16(-7)^2 - 625 = 784 - 625 = 159 > 0 \Rightarrow \text{True}
\][/tex]
8. Conclusion:
Jerald is less than 104 feet above the ground for [tex]\(t > 6.25\)[/tex].
Thus, the correct answer is:
[tex]\[ t > 6.25 \][/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to determine when this height is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this inequality step-by-step:
1. Subtract 104 from both sides:
[tex]\[
-16t^2 + 729 - 104 < 0
\][/tex]
This simplifies to:
[tex]\[
-16t^2 + 625 < 0
\][/tex]
2. Rearrange the terms:
[tex]\[
625 - 16t^2 < 0
\][/tex]
3. Multiply everything by -1 (this reverses the inequality):
[tex]\[
16t^2 - 625 > 0
\][/tex]
4. Factor the quadratic expression:
[tex]\[
(4t - 25)(4t + 25) > 0
\][/tex]
5. Find the critical points by solving [tex]\(4t - 25 = 0\)[/tex] and [tex]\(4t + 25 = 0\)[/tex]:
[tex]\[
4t - 25 = 0 \Rightarrow t = \frac{25}{4} = 6.25
\][/tex]
[tex]\[
4t + 25 = 0 \Rightarrow t = -\frac{25}{4} = -6.25
\][/tex]
6. Determine the intervals to test the inequality:
The critical points divide the number line into three intervals:
[tex]\[
(-\infty, -6.25), \quad (-6.25, 6.25), \quad (6.25, \infty)
\][/tex]
7. Test a point from each interval to see where the inequality holds:
- Interval [tex]\((-6.25, 6.25)\)[/tex]:
Choose [tex]\(t = 0\)[/tex]:
[tex]\[
16(0)^2 - 625 = -625 \Rightarrow \text{False}
\][/tex]
- Interval [tex]\((6.25, \infty)\)[/tex]:
Choose [tex]\(t = 7\)[/tex]:
[tex]\[
16(7)^2 - 625 = 784 - 625 = 159 > 0 \Rightarrow \text{True}
\][/tex]
- Interval [tex]\((-\infty, -6.25)\)[/tex]:
Choose [tex]\(t = -7\)[/tex]:
[tex]\[
16(-7)^2 - 625 = 784 - 625 = 159 > 0 \Rightarrow \text{True}
\][/tex]
8. Conclusion:
Jerald is less than 104 feet above the ground for [tex]\(t > 6.25\)[/tex].
Thus, the correct answer is:
[tex]\[ t > 6.25 \][/tex]
