Answer :
To find the interval of time during which Jerald is less than 104 feet above the ground, we need to solve the inequality based on the given height equation:
The equation for Jerald's height is:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find the values of [tex]\( t \)[/tex] where his height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
### Step-by-step solution:
1. Set up the inequality:
[tex]\[-16t^2 + 729 < 104\][/tex]
2. Simplify the inequality:
- First, subtract 729 from both sides to isolate the term with [tex]\( t^2 \)[/tex]:
[tex]\[-16t^2 < 104 - 729\][/tex]
- Calculate [tex]\( 104 - 729 \)[/tex]:
[tex]\[-16t^2 < -625\][/tex]
3. Divide the inequality by -16:
- Since we're dividing by a negative number, the direction of the inequality changes:
[tex]\[t^2 > \frac{625}{16}\][/tex]
4. Simplify further:
- Calculate [tex]\(\frac{625}{16}\)[/tex]:
[tex]\[t^2 > 39.0625\][/tex]
5. Solve for [tex]\( t \)[/tex]:
- Take the square root of both sides:
[tex]\[t > \sqrt{39.0625}\][/tex]
[tex]\[t < -\sqrt{39.0625}\][/tex]
- Calculate the square root of 39.0625:
[tex]\[\sqrt{39.0625} = 6.25\][/tex]
6. Determine interval:
- From the inequality [tex]\( t^2 > 39.0625 \)[/tex], we have two intervals:
- [tex]\( t > 6.25 \)[/tex] and [tex]\( t < -6.25 \)[/tex]
- Since time [tex]\( t \)[/tex] represents seconds and cannot be negative in this context, you only consider the positive interval:
- [tex]\( t > 6.25 \)[/tex]
Therefore, Jerald is less than 104 feet above the ground for [tex]\( t > 6.25 \)[/tex] seconds.
The equation for Jerald's height is:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find the values of [tex]\( t \)[/tex] where his height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
### Step-by-step solution:
1. Set up the inequality:
[tex]\[-16t^2 + 729 < 104\][/tex]
2. Simplify the inequality:
- First, subtract 729 from both sides to isolate the term with [tex]\( t^2 \)[/tex]:
[tex]\[-16t^2 < 104 - 729\][/tex]
- Calculate [tex]\( 104 - 729 \)[/tex]:
[tex]\[-16t^2 < -625\][/tex]
3. Divide the inequality by -16:
- Since we're dividing by a negative number, the direction of the inequality changes:
[tex]\[t^2 > \frac{625}{16}\][/tex]
4. Simplify further:
- Calculate [tex]\(\frac{625}{16}\)[/tex]:
[tex]\[t^2 > 39.0625\][/tex]
5. Solve for [tex]\( t \)[/tex]:
- Take the square root of both sides:
[tex]\[t > \sqrt{39.0625}\][/tex]
[tex]\[t < -\sqrt{39.0625}\][/tex]
- Calculate the square root of 39.0625:
[tex]\[\sqrt{39.0625} = 6.25\][/tex]
6. Determine interval:
- From the inequality [tex]\( t^2 > 39.0625 \)[/tex], we have two intervals:
- [tex]\( t > 6.25 \)[/tex] and [tex]\( t < -6.25 \)[/tex]
- Since time [tex]\( t \)[/tex] represents seconds and cannot be negative in this context, you only consider the positive interval:
- [tex]\( t > 6.25 \)[/tex]
Therefore, Jerald is less than 104 feet above the ground for [tex]\( t > 6.25 \)[/tex] seconds.
