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Answer :
To determine the interval of time where Jerald is less than 104 feet above the ground, we start with the equation that models his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when his height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's rearrange the inequality:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Next, divide both sides by -16, remembering to reverse the inequality sign because we're dividing by a negative number:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Now, solve for [tex]\( t \)[/tex] by taking the square root of both sides:
[tex]\[ t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}} \][/tex]
Calculating that gives:
[tex]\[ t > \frac{25}{4} \quad \text{or} \quad t < -\frac{25}{4} \][/tex]
Simplifying the values:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Since time [tex]\( t \)[/tex] cannot be negative in this context, we only consider positive values of [tex]\( t \)[/tex]. Therefore, the interval where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
So, the correct choice is [tex]\( t > 6.25 \)[/tex].
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when his height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, let's rearrange the inequality:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Next, divide both sides by -16, remembering to reverse the inequality sign because we're dividing by a negative number:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Now, solve for [tex]\( t \)[/tex] by taking the square root of both sides:
[tex]\[ t > \sqrt{\frac{625}{16}} \quad \text{or} \quad t < -\sqrt{\frac{625}{16}} \][/tex]
Calculating that gives:
[tex]\[ t > \frac{25}{4} \quad \text{or} \quad t < -\frac{25}{4} \][/tex]
Simplifying the values:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Since time [tex]\( t \)[/tex] cannot be negative in this context, we only consider positive values of [tex]\( t \)[/tex]. Therefore, the interval where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
So, the correct choice is [tex]\( t > 6.25 \)[/tex].
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