Answer :
To determine the interval of time for which Jerald is less than 104 feet above the ground based on the height equation [tex]\( h = -t^2 + 729 \)[/tex], we need to find the values of [tex]\( t \)[/tex] where the height [tex]\( h \)[/tex] is less than 104 feet.
1. Set up the inequality:
[tex]\[
-t^2 + 729 < 104
\][/tex]
2. Solve for [tex]\( t \)[/tex] to find the points where the height is exactly 104 feet:
[tex]\[
-t^2 + 729 = 104
\][/tex]
3. Simplify the equation to isolate [tex]\( t^2 \)[/tex]:
[tex]\[
-t^2 + 729 = 104
\][/tex]
[tex]\[
729 - 104 = t^2
\][/tex]
[tex]\[
625 = t^2
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[
t^2 = 625
\][/tex]
[tex]\[
t = \pm 25
\][/tex]
So, the values of [tex]\( t \)[/tex] are [tex]\( t = -25 \)[/tex] and [tex]\( t = 25 \)[/tex].
5. Determine the interval where the height is less than 104 feet:
Notice here that the height function [tex]\( h = -t^2 + 729 \)[/tex] is a downward-opening parabola. This means that Jerald's height will be greater than or equal to 104 feet when [tex]\( -25 \leq t \leq 25 \)[/tex], and it will be less than 104 feet outside this interval.
Therefore, Jerald is less than 104 feet above the ground for the interval:
[tex]\[
t < -25 \quad \text{or} \quad t > 25
\][/tex]
Given the context of the problem (a real-world scenario of time and height), negative values for [tex]\( t \)[/tex] generally don't make sense (since [tex]\( t \)[/tex] represents time). Thus, we focus on the positive values:
For practical purposes, the interval of time where Jerald is less than 104 feet above the ground is:
[tex]\[
t > 25
\][/tex]
1. Set up the inequality:
[tex]\[
-t^2 + 729 < 104
\][/tex]
2. Solve for [tex]\( t \)[/tex] to find the points where the height is exactly 104 feet:
[tex]\[
-t^2 + 729 = 104
\][/tex]
3. Simplify the equation to isolate [tex]\( t^2 \)[/tex]:
[tex]\[
-t^2 + 729 = 104
\][/tex]
[tex]\[
729 - 104 = t^2
\][/tex]
[tex]\[
625 = t^2
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[
t^2 = 625
\][/tex]
[tex]\[
t = \pm 25
\][/tex]
So, the values of [tex]\( t \)[/tex] are [tex]\( t = -25 \)[/tex] and [tex]\( t = 25 \)[/tex].
5. Determine the interval where the height is less than 104 feet:
Notice here that the height function [tex]\( h = -t^2 + 729 \)[/tex] is a downward-opening parabola. This means that Jerald's height will be greater than or equal to 104 feet when [tex]\( -25 \leq t \leq 25 \)[/tex], and it will be less than 104 feet outside this interval.
Therefore, Jerald is less than 104 feet above the ground for the interval:
[tex]\[
t < -25 \quad \text{or} \quad t > 25
\][/tex]
Given the context of the problem (a real-world scenario of time and height), negative values for [tex]\( t \)[/tex] generally don't make sense (since [tex]\( t \)[/tex] represents time). Thus, we focus on the positive values:
For practical purposes, the interval of time where Jerald is less than 104 feet above the ground is:
[tex]\[
t > 25
\][/tex]
