Answer :
To solve this problem, we need to find the interval of time [tex]\( t \)[/tex] during which Jerald is less than 104 feet above the ground.
The height of Jerald as a function of time is given by the equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need his height [tex]\( h \)[/tex] to be less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this step-by-step:
1. Subtract 729 from both sides to isolate the [tex]\( t \)[/tex]-terms:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
2. Divide both sides by -16. Remember that dividing by a negative number reverses the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
3. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{39.0625} \quad \text{or} \quad t < -\sqrt{39.0625} \][/tex]
Calculating the square root of 39.0625, we find:
[tex]\[ \sqrt{39.0625} = 6.25 \][/tex]
This means that the solutions for [tex]\( t \)[/tex] are:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
However, since the context of the problem indicates that time [tex]\( t \)[/tex] cannot be negative (as we're considering a real-world scenario starting from [tex]\( t=0 \)[/tex]), we only consider positive time intervals. Therefore, we disregard [tex]\( t < -6.25 \)[/tex].
The correct interval for time when Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
Thus, from the given options, the correct answer is:
[tex]\[ t > 6.25 \][/tex]
The height of Jerald as a function of time is given by the equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need his height [tex]\( h \)[/tex] to be less than 104 feet, so we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Let's solve this step-by-step:
1. Subtract 729 from both sides to isolate the [tex]\( t \)[/tex]-terms:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
2. Divide both sides by -16. Remember that dividing by a negative number reverses the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
3. Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t > \sqrt{39.0625} \quad \text{or} \quad t < -\sqrt{39.0625} \][/tex]
Calculating the square root of 39.0625, we find:
[tex]\[ \sqrt{39.0625} = 6.25 \][/tex]
This means that the solutions for [tex]\( t \)[/tex] are:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
However, since the context of the problem indicates that time [tex]\( t \)[/tex] cannot be negative (as we're considering a real-world scenario starting from [tex]\( t=0 \)[/tex]), we only consider positive time intervals. Therefore, we disregard [tex]\( t < -6.25 \)[/tex].
The correct interval for time when Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
Thus, from the given options, the correct answer is:
[tex]\[ t > 6.25 \][/tex]
