Answer :
To find the interval of time during which Jerald's height is less than 104 feet above the ground, we start with the equation that represents his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when his height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Next, we'll solve this inequality step by step:
1. Subtract 729 from both sides to isolate the term with [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 + 729 - 729 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
2. Divide both sides by -16. Remember that when you divide or multiply an inequality by a negative number, the inequality sign flips:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
3. Calculate [tex]\( \frac{625}{16} \)[/tex]:
[tex]\[ \frac{625}{16} = 39.0625 \][/tex]
4. 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:
[tex]\[ \sqrt{39.0625} = 6.25 \][/tex]
So, the inequality becomes:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Since time cannot be negative when measuring from the start of the jump, we only consider the positive form:
[tex]\[ t > 6.25 \][/tex]
Thus, Jerald is less than 104 feet above the ground for the interval [tex]\( t > 6.25 \)[/tex].
Therefore, the correct interval of time is:
[tex]\[ t > 6.25 \][/tex]
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when his height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Next, we'll solve this inequality step by step:
1. Subtract 729 from both sides to isolate the term with [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 + 729 - 729 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
2. Divide both sides by -16. Remember that when you divide or multiply an inequality by a negative number, the inequality sign flips:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
3. Calculate [tex]\( \frac{625}{16} \)[/tex]:
[tex]\[ \frac{625}{16} = 39.0625 \][/tex]
4. 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:
[tex]\[ \sqrt{39.0625} = 6.25 \][/tex]
So, the inequality becomes:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Since time cannot be negative when measuring from the start of the jump, we only consider the positive form:
[tex]\[ t > 6.25 \][/tex]
Thus, Jerald is less than 104 feet above the ground for the interval [tex]\( t > 6.25 \)[/tex].
Therefore, the correct interval of time is:
[tex]\[ t > 6.25 \][/tex]
