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Answer :
We are given the height equation
[tex]$$
h = -16t^2 + 729,
$$[/tex]
and we want to find the time interval when the height is less than 104 feet, i.e., when
[tex]$$
-16t^2 + 729 < 104.
$$[/tex]
A good first step is to find when the height is exactly 104 feet. Set
[tex]$$
-16t^2 + 729 = 104.
$$[/tex]
Subtract 104 from both sides to isolate the [tex]$t^2$[/tex] term:
[tex]$$
-16t^2 = 104 - 729.
$$[/tex]
Simplify the right side:
[tex]$$
-16t^2 = -625.
$$[/tex]
Divide both sides by [tex]$-16$[/tex] (noting that dividing by a negative number will flip the sign):
[tex]$$
t^2 = \frac{625}{16}.
$$[/tex]
Taking the square root of both sides gives
[tex]$$
t = \sqrt{\frac{625}{16}} = \frac{25}{4} = 6.25.
$$[/tex]
Since [tex]$t$[/tex] represents time, we consider only the positive solution ([tex]$t \ge 0$[/tex]).
Next, we analyze the inequality. The height equation represents a downward-opening parabola (because the coefficient of [tex]$t^2$[/tex] is negative). This means that for times before [tex]$t = 6.25$[/tex] seconds, Jerald’s height is above 104 feet, and after [tex]$t = 6.25$[/tex] seconds, his height falls below 104 feet.
Thus, Jerald is less than 104 feet above the ground when
[tex]$$
t > 6.25.
$$[/tex]
The correct interval, considering the domain of time ([tex]$t \ge 0$[/tex]), is
[tex]$$
t > 6.25.
$$[/tex]
[tex]$$
h = -16t^2 + 729,
$$[/tex]
and we want to find the time interval when the height is less than 104 feet, i.e., when
[tex]$$
-16t^2 + 729 < 104.
$$[/tex]
A good first step is to find when the height is exactly 104 feet. Set
[tex]$$
-16t^2 + 729 = 104.
$$[/tex]
Subtract 104 from both sides to isolate the [tex]$t^2$[/tex] term:
[tex]$$
-16t^2 = 104 - 729.
$$[/tex]
Simplify the right side:
[tex]$$
-16t^2 = -625.
$$[/tex]
Divide both sides by [tex]$-16$[/tex] (noting that dividing by a negative number will flip the sign):
[tex]$$
t^2 = \frac{625}{16}.
$$[/tex]
Taking the square root of both sides gives
[tex]$$
t = \sqrt{\frac{625}{16}} = \frac{25}{4} = 6.25.
$$[/tex]
Since [tex]$t$[/tex] represents time, we consider only the positive solution ([tex]$t \ge 0$[/tex]).
Next, we analyze the inequality. The height equation represents a downward-opening parabola (because the coefficient of [tex]$t^2$[/tex] is negative). This means that for times before [tex]$t = 6.25$[/tex] seconds, Jerald’s height is above 104 feet, and after [tex]$t = 6.25$[/tex] seconds, his height falls below 104 feet.
Thus, Jerald is less than 104 feet above the ground when
[tex]$$
t > 6.25.
$$[/tex]
The correct interval, considering the domain of time ([tex]$t \ge 0$[/tex]), is
[tex]$$
t > 6.25.
$$[/tex]
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