Answer :
Sure, let's solve this problem step-by-step to find the interval of time during which Jerald is less than 104 feet above the ground.
Given the height equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step 1: Subtract 104 from both sides to form a standard inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
Step 2: Simplify and solve for [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 < -625 \][/tex]
Divide both sides by -16 (note: this reverses the inequality sign):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
Step 3: Take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
So, the time [tex]\( t \)[/tex] must be greater than 6.25 seconds for Jerald to be less than 104 feet above the ground.
Now, let's match this result to one of the provided intervals:
1. [tex]\( t > 6.25 \)[/tex]
2. [tex]\( -6.25 < t < 6.25 \)[/tex]
3. [tex]\( t < 6.25 \)[/tex]
4. [tex]\( 0 \leq t \leq 6.25 \)[/tex]
From our solution, the interval where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
Therefore, the correct answer is:
[tex]\[ t > 6.25 \][/tex]
Given the height equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We need to determine when:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Step 1: Subtract 104 from both sides to form a standard inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
Step 2: Simplify and solve for [tex]\( t^2 \)[/tex]:
[tex]\[ -16t^2 < -625 \][/tex]
Divide both sides by -16 (note: this reverses the inequality sign):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
Step 3: Take the square root of both sides:
[tex]\[ t > \sqrt{39.0625} \][/tex]
[tex]\[ t > 6.25 \][/tex]
So, the time [tex]\( t \)[/tex] must be greater than 6.25 seconds for Jerald to be less than 104 feet above the ground.
Now, let's match this result to one of the provided intervals:
1. [tex]\( t > 6.25 \)[/tex]
2. [tex]\( -6.25 < t < 6.25 \)[/tex]
3. [tex]\( t < 6.25 \)[/tex]
4. [tex]\( 0 \leq t \leq 6.25 \)[/tex]
From our solution, the interval where Jerald is less than 104 feet above the ground is:
[tex]\[ t > 6.25 \][/tex]
Therefore, the correct answer is:
[tex]\[ t > 6.25 \][/tex]