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Answer :
Sure! Let's solve the problem step-by-step.
The equation that models Jerald's height in feet as he jumps is given by:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine for which interval of time [tex]\( t \)[/tex], his height is less than 104 feet. Thus, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's simplify the inequality:
1. Subtract 729 from both sides:
[tex]\[-16t^2 < 104 - 729\][/tex]
2. Calculate the right side:
[tex]\[-16t^2 < -625\][/tex]
3. Divide both sides by -16. Remember, dividing or multiplying an inequality by a negative number flips the inequality sign:
[tex]\[t^2 > \frac{625}{16}\][/tex]
4. Find the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \text{ or } t < -\sqrt{\frac{625}{16}} \][/tex]
Calculate [tex]\( \sqrt{\frac{625}{16}} \)[/tex] which is [tex]\(\frac{25}{4}\)[/tex] or 6.25.
Therefore, [tex]\( t > 6.25 \text{ or } t < -6.25 \)[/tex].
Since time, [tex]\( t \)[/tex], must be non-negative (because it's not possible to go back in time for this context), we only consider the positive interval:
Thus, the interval of time for which Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
So, the correct answer is [tex]\( t > 6.25 \)[/tex].
The equation that models Jerald's height in feet as he jumps is given by:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine for which interval of time [tex]\( t \)[/tex], his height is less than 104 feet. Thus, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's simplify the inequality:
1. Subtract 729 from both sides:
[tex]\[-16t^2 < 104 - 729\][/tex]
2. Calculate the right side:
[tex]\[-16t^2 < -625\][/tex]
3. Divide both sides by -16. Remember, dividing or multiplying an inequality by a negative number flips the inequality sign:
[tex]\[t^2 > \frac{625}{16}\][/tex]
4. Find the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \text{ or } t < -\sqrt{\frac{625}{16}} \][/tex]
Calculate [tex]\( \sqrt{\frac{625}{16}} \)[/tex] which is [tex]\(\frac{25}{4}\)[/tex] or 6.25.
Therefore, [tex]\( t > 6.25 \text{ or } t < -6.25 \)[/tex].
Since time, [tex]\( t \)[/tex], must be non-negative (because it's not possible to go back in time for this context), we only consider the positive interval:
Thus, the interval of time for which Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
So, the correct answer is [tex]\( t > 6.25 \)[/tex].
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