Answer :
To determine the interval of time during which Jerald is less than 104 feet above the ground, we'll use the given equation for his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
Here, [tex]\( h \)[/tex] represents Jerald's height in feet, and [tex]\( t \)[/tex] represents the time in seconds.
The problem asks when his height is less than 104 feet. So, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this step by step:
1. Subtract 104 from both sides to simplify the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Subtract 625 from both sides:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide everything by -16: Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
Since time cannot be negative and we are interested in the time after Jerald first falls below 104 feet, the interval where Jerald is less than 104 feet above the ground is when [tex]\( t \)[/tex] is greater than 6.25 seconds. So the interval is:
[tex]\[ t > 6.25 \][/tex]
In conclusion, the correct answer from the options provided is:
[tex]\( t > 6.25 \)[/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
Here, [tex]\( h \)[/tex] represents Jerald's height in feet, and [tex]\( t \)[/tex] represents the time in seconds.
The problem asks when his height is less than 104 feet. So, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this step by step:
1. Subtract 104 from both sides to simplify the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Subtract 625 from both sides:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide everything by -16: Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
[tex]\[ t > \frac{25}{4} \][/tex]
[tex]\[ t > 6.25 \][/tex]
Since time cannot be negative and we are interested in the time after Jerald first falls below 104 feet, the interval where Jerald is less than 104 feet above the ground is when [tex]\( t \)[/tex] is greater than 6.25 seconds. So the interval is:
[tex]\[ t > 6.25 \][/tex]
In conclusion, the correct answer from the options provided is:
[tex]\( t > 6.25 \)[/tex]
