Answer :
Sure! Let's solve the problem step-by-step to understand for which interval of time Jerald is less than 104 feet above the ground.
We are given the equation for Jerald's height as [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( t \)[/tex] is the time in seconds. We need to find the time interval when his height [tex]\( h \)[/tex] is less than 104 feet.
1. Set up the inequality based on the condition that his height is less than 104 feet:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the inequality to isolate the quadratic term:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
[tex]\[
-16t^2 < -625
\][/tex]
3. Divide both sides of the inequality by [tex]\(-16\)[/tex]. Remember, when dividing by a negative number, the inequality sign flips:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. 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]
5. Calculate the square root:
[tex]\[
\sqrt{\frac{625}{16}} = \frac{\sqrt{625}}{\sqrt{16}} = \frac{25}{4} = 6.25
\][/tex]
6. Based on the calculation, we have two conditions [tex]\( t > 6.25 \)[/tex] or [tex]\( t < -6.25 \)[/tex]. However, because time cannot be negative, we discard [tex]\( t < -6.25 \)[/tex].
Therefore, the interval during which Jerald is less than 104 feet above the ground is [tex]\( t > 6.25 \)[/tex] seconds.
We are given the equation for Jerald's height as [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( t \)[/tex] is the time in seconds. We need to find the time interval when his height [tex]\( h \)[/tex] is less than 104 feet.
1. Set up the inequality based on the condition that his height is less than 104 feet:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the inequality to isolate the quadratic term:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
[tex]\[
-16t^2 < -625
\][/tex]
3. Divide both sides of the inequality by [tex]\(-16\)[/tex]. Remember, when dividing by a negative number, the inequality sign flips:
[tex]\[
t^2 > \frac{625}{16}
\][/tex]
4. 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]
5. Calculate the square root:
[tex]\[
\sqrt{\frac{625}{16}} = \frac{\sqrt{625}}{\sqrt{16}} = \frac{25}{4} = 6.25
\][/tex]
6. Based on the calculation, we have two conditions [tex]\( t > 6.25 \)[/tex] or [tex]\( t < -6.25 \)[/tex]. However, because time cannot be negative, we discard [tex]\( t < -6.25 \)[/tex].
Therefore, the interval during which Jerald is less than 104 feet above the ground is [tex]\( t > 6.25 \)[/tex] seconds.
