Answer :
To determine for which interval of time Jerald is less than 104 feet above the ground, we start with the given equation for his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when his height is less than 104 feet, so we'll set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Rearrange the inequality to isolate the term with [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Next, we divide both sides by -16. Remember that dividing an inequality by a negative number reverses the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Now, calculate the value:
[tex]\[ \frac{625}{16} = 39.0625 \][/tex]
So, we have:
[tex]\[ t^2 > 39.0625 \][/tex]
To solve for [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t < -\sqrt{39.0625} \][/tex]
Calculate the square roots:
[tex]\[ \sqrt{39.0625} = 6.25 \][/tex]
Since [tex]\( t \)[/tex] represents time, and time cannot be negative in this context, we only consider the positive value:
The valid interval for [tex]\( t \)[/tex] is [tex]\( t > 6.25 \)[/tex].
So, Jerald is less than 104 feet above the ground for [tex]\( t > 6.25 \)[/tex] seconds.
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when his height is less than 104 feet, so we'll set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Rearrange the inequality to isolate the term with [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Next, we divide both sides by -16. Remember that dividing an inequality by a negative number reverses the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Now, calculate the value:
[tex]\[ \frac{625}{16} = 39.0625 \][/tex]
So, we have:
[tex]\[ t^2 > 39.0625 \][/tex]
To solve for [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t < -\sqrt{39.0625} \][/tex]
Calculate the square roots:
[tex]\[ \sqrt{39.0625} = 6.25 \][/tex]
Since [tex]\( t \)[/tex] represents time, and time cannot be negative in this context, we only consider the positive value:
The valid interval for [tex]\( t \)[/tex] is [tex]\( t > 6.25 \)[/tex].
So, Jerald is less than 104 feet above the ground for [tex]\( t > 6.25 \)[/tex] seconds.