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Answer :
To solve the problem of finding when Jerald's height is less than 104 feet, given the equation [tex]\( h = -16t^2 + 729 \)[/tex], we need to determine for which values of [tex]\( t \)[/tex] the height [tex]\( h \)[/tex] is less than 104 feet.
1. Start with the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Subtract 104 from both sides to set up the inequality:
[tex]\[
-16t^2 + 729 - 104 < 0
\][/tex]
3. Simplify the expression:
[tex]\[
-16t^2 + 625 < 0
\][/tex]
4. Rearrange the terms for clarity:
[tex]\[
625 - 16t^2 < 0
\][/tex]
5. Move the 625 to the other side:
[tex]\[
-16t^2 < -625
\][/tex]
6. Divide every term by -16. Remember that dividing by a negative number flips the inequality sign:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
7. Simplify the fraction:
[tex]\[
t^2 > \left(\frac{25}{4}\right)^2
\][/tex]
8. Take the square root of both sides. Since we are dealing with a squared term, we consider both the positive and negative roots:
[tex]\[
t < -\frac{25}{4} \quad \text{or} \quad t > \frac{25}{4}
\][/tex]
Thus, the interval of time during which Jerald's height is less than 104 feet is when [tex]\( t \)[/tex] is less than [tex]\(-6.25\)[/tex] or [tex]\( t \)[/tex] is greater than [tex]\(6.25\)[/tex]. Since time [tex]\( t \)[/tex] represents seconds and must be a non-negative value (as Jerald cannot jump before time zero), we only consider the positive interval:
[tex]\( t > 6.25 \)[/tex]
Therefore, the correct answer is:
- [tex]\( t > 6.25 \)[/tex]
1. Start with the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Subtract 104 from both sides to set up the inequality:
[tex]\[
-16t^2 + 729 - 104 < 0
\][/tex]
3. Simplify the expression:
[tex]\[
-16t^2 + 625 < 0
\][/tex]
4. Rearrange the terms for clarity:
[tex]\[
625 - 16t^2 < 0
\][/tex]
5. Move the 625 to the other side:
[tex]\[
-16t^2 < -625
\][/tex]
6. Divide every term by -16. Remember that dividing by a negative number flips the inequality sign:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
7. Simplify the fraction:
[tex]\[
t^2 > \left(\frac{25}{4}\right)^2
\][/tex]
8. Take the square root of both sides. Since we are dealing with a squared term, we consider both the positive and negative roots:
[tex]\[
t < -\frac{25}{4} \quad \text{or} \quad t > \frac{25}{4}
\][/tex]
Thus, the interval of time during which Jerald's height is less than 104 feet is when [tex]\( t \)[/tex] is less than [tex]\(-6.25\)[/tex] or [tex]\( t \)[/tex] is greater than [tex]\(6.25\)[/tex]. Since time [tex]\( t \)[/tex] represents seconds and must be a non-negative value (as Jerald cannot jump before time zero), we only consider the positive interval:
[tex]\( t > 6.25 \)[/tex]
Therefore, the correct answer is:
- [tex]\( t > 6.25 \)[/tex]
Thanks for taking the time to read Jerald jumped from a bungee tower The equation that models his height in feet is tex h 16t 2 729 tex where tex t tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
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