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Answer :
We start with the equation for height:
[tex]$$
h = -16t^2 + 729
$$[/tex]
and we want to find for which time interval the height is less than 104 feet. That is, we need to solve:
[tex]$$
-16t^2 + 729 < 104.
$$[/tex]
A good first step is to find the time when the height exactly equals 104 feet. So we set:
[tex]$$
-16t^2 + 729 = 104.
$$[/tex]
Subtract 729 from both sides to isolate the term with [tex]$t^2$[/tex]:
[tex]$$
-16t^2 = 104 - 729.
$$[/tex]
Simplify the right-hand side:
[tex]$$
104 - 729 = -625,
$$[/tex]
so we have:
[tex]$$
-16t^2 = -625.
$$[/tex]
Next, divide both sides by [tex]$-16$[/tex]:
[tex]$$
t^2 = \frac{625}{16}.
$$[/tex]
Taking the square root of both sides gives:
[tex]$$
t = \pm \frac{25}{4}.
$$[/tex]
Since time cannot be negative, we choose the positive value:
[tex]$$
t = \frac{25}{4} = 6.25.
$$[/tex]
At [tex]$t = 6.25$[/tex] seconds, Jerald’s height is exactly 104 feet. To determine the interval where his height is less than 104 feet, we analyze the behavior of the quadratic function. The function
[tex]$$
h = -16t^2 + 729
$$[/tex]
is a downward-opening parabola. This means that before reaching 104 feet on the way down, his height is greater than 104 feet. Once he passes the point [tex]$t = 6.25$[/tex] seconds, his height drops below 104 feet.
Therefore, for times greater than [tex]$6.25$[/tex] seconds, his height [tex]$h$[/tex] satisfies:
[tex]$$
h < 104.
$$[/tex]
Thus, the interval of time during which Jerald is less than 104 feet above the ground is:
[tex]$$
t > 6.25.
$$[/tex]
[tex]$$
h = -16t^2 + 729
$$[/tex]
and we want to find for which time interval the height is less than 104 feet. That is, we need to solve:
[tex]$$
-16t^2 + 729 < 104.
$$[/tex]
A good first step is to find the time when the height exactly equals 104 feet. So we set:
[tex]$$
-16t^2 + 729 = 104.
$$[/tex]
Subtract 729 from both sides to isolate the term with [tex]$t^2$[/tex]:
[tex]$$
-16t^2 = 104 - 729.
$$[/tex]
Simplify the right-hand side:
[tex]$$
104 - 729 = -625,
$$[/tex]
so we have:
[tex]$$
-16t^2 = -625.
$$[/tex]
Next, divide both sides by [tex]$-16$[/tex]:
[tex]$$
t^2 = \frac{625}{16}.
$$[/tex]
Taking the square root of both sides gives:
[tex]$$
t = \pm \frac{25}{4}.
$$[/tex]
Since time cannot be negative, we choose the positive value:
[tex]$$
t = \frac{25}{4} = 6.25.
$$[/tex]
At [tex]$t = 6.25$[/tex] seconds, Jerald’s height is exactly 104 feet. To determine the interval where his height is less than 104 feet, we analyze the behavior of the quadratic function. The function
[tex]$$
h = -16t^2 + 729
$$[/tex]
is a downward-opening parabola. This means that before reaching 104 feet on the way down, his height is greater than 104 feet. Once he passes the point [tex]$t = 6.25$[/tex] seconds, his height drops below 104 feet.
Therefore, for times greater than [tex]$6.25$[/tex] seconds, his height [tex]$h$[/tex] satisfies:
[tex]$$
h < 104.
$$[/tex]
Thus, the interval of time during which Jerald is less than 104 feet above the ground is:
[tex]$$
t > 6.25.
$$[/tex]
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