Answer :
Let's solve the problem step-by-step to find for which interval of time Jerald is less than 104 feet above the ground.
1. Understand the equation:
The equation given is [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( h \)[/tex] is Jerald's height in feet and [tex]\( t \)[/tex] is the time in seconds.
2. Set up the inequality:
We are asked to find when Jerald's height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
3. Simplify the inequality:
First, subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
4. Rearrange the inequality:
Now, move the constant to the other side to isolate [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 < -625 \][/tex]
Dividing both sides by -16 (remember to reverse the inequality because you are dividing by a negative number):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
5. Solve for [tex]\( t \)[/tex]:
Take the square root of both sides:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
Calculating the square root:
[tex]\[ t > \frac{25}{4} \][/tex]
Therefore, the critical value where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
6. Conclusion:
Jerald is less than 104 feet above the ground when time [tex]\( t \)[/tex] is greater than 6.25 seconds. This corresponds to the interval [tex]\( t > 6.25 \)[/tex].
So, the correct answer is:
[tex]\[ t > 6.25 \][/tex]
1. Understand the equation:
The equation given is [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( h \)[/tex] is Jerald's height in feet and [tex]\( t \)[/tex] is the time in seconds.
2. Set up the inequality:
We are asked to find when Jerald's height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
3. Simplify the inequality:
First, subtract 104 from both sides:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
4. Rearrange the inequality:
Now, move the constant to the other side to isolate [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 < -625 \][/tex]
Dividing both sides by -16 (remember to reverse the inequality because you are dividing by a negative number):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
5. Solve for [tex]\( t \)[/tex]:
Take the square root of both sides:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
Calculating the square root:
[tex]\[ t > \frac{25}{4} \][/tex]
Therefore, the critical value where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
6. Conclusion:
Jerald is less than 104 feet above the ground when time [tex]\( t \)[/tex] is greater than 6.25 seconds. This corresponds to the interval [tex]\( t > 6.25 \)[/tex].
So, the correct answer is:
[tex]\[ t > 6.25 \][/tex]
