Answer :
To solve this problem, we want to find when Jerald's height is less than 104 feet. The height equation is given by [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( t \)[/tex] is the time in seconds.
We're looking for the time intervals when [tex]\( h < 104 \)[/tex].
### Step-by-step Solution:
1. Set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Isolate the quadratic term:
First, subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
This simplifies to:
[tex]\[
-16t^2 < -625
\][/tex]
3. Divide by -16:
We need to divide both sides by -16, and remember that dividing or multiplying by a negative number reverses the inequality:
[tex]\[
t^2 > \frac{-625}{-16}
\][/tex]
Simplifying further gives:
[tex]\[
t^2 > 39.0625
\][/tex]
4. Find the square root of both sides:
To solve for [tex]\( t \)[/tex], take the square root:
[tex]\[
t > \sqrt{39.0625}
\][/tex]
Which gives:
[tex]\[
t > 6.25
\][/tex]
However, we also need to consider the square root of a negative result as:
[tex]\[
t < -\sqrt{39.0625}, \text{ but negative time does not make sense here.}
\][/tex]
So we discard the negative value.
Since time [tex]\( t \)[/tex] must be positive, the valid interval is [tex]\( t > 6.25 \)[/tex].
Therefore, Jerald's height is less than 104 feet above the ground for [tex]\( t > 6.25 \)[/tex] seconds. The correct interval is [tex]\( t > 6.25 \)[/tex].
We're looking for the time intervals when [tex]\( h < 104 \)[/tex].
### Step-by-step Solution:
1. Set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Isolate the quadratic term:
First, subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
This simplifies to:
[tex]\[
-16t^2 < -625
\][/tex]
3. Divide by -16:
We need to divide both sides by -16, and remember that dividing or multiplying by a negative number reverses the inequality:
[tex]\[
t^2 > \frac{-625}{-16}
\][/tex]
Simplifying further gives:
[tex]\[
t^2 > 39.0625
\][/tex]
4. Find the square root of both sides:
To solve for [tex]\( t \)[/tex], take the square root:
[tex]\[
t > \sqrt{39.0625}
\][/tex]
Which gives:
[tex]\[
t > 6.25
\][/tex]
However, we also need to consider the square root of a negative result as:
[tex]\[
t < -\sqrt{39.0625}, \text{ but negative time does not make sense here.}
\][/tex]
So we discard the negative value.
Since time [tex]\( t \)[/tex] must be positive, the valid interval is [tex]\( t > 6.25 \)[/tex].
Therefore, Jerald's height is less than 104 feet above the ground for [tex]\( t > 6.25 \)[/tex] seconds. The correct interval is [tex]\( t > 6.25 \)[/tex].
