Answer :
To determine the interval for which Jerald is less than 104 feet above the ground, let's start with the given height equation:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find the time [tex]\( t \)[/tex] where Jerald’s height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, subtract 104 from both sides to move all terms to one side of the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
Next, isolate the quadratic term:
[tex]\[ -16t^2 < -625 \][/tex]
Divide both sides by -16 (remember, dividing by a negative number reverses the inequality sign):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
To solve for [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[ |t| > \sqrt{39.0625} \][/tex]
[tex]\[ |t| > 6.25 \][/tex]
This inequality tells us that [tex]\( t \)[/tex] can be any value greater than 6.25 seconds or less than -6.25 seconds. Since time [tex]\( t \)[/tex] in this context is usually considered non-negative (Jerald cannot go back in time), we discard the negative interval.
Thus, the interval of time for which 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]
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find the time [tex]\( t \)[/tex] where Jerald’s height [tex]\( h \)[/tex] is less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, subtract 104 from both sides to move all terms to one side of the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
Next, isolate the quadratic term:
[tex]\[ -16t^2 < -625 \][/tex]
Divide both sides by -16 (remember, dividing by a negative number reverses the inequality sign):
[tex]\[ t^2 > \frac{625}{16} \][/tex]
[tex]\[ t^2 > 39.0625 \][/tex]
To solve for [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[ |t| > \sqrt{39.0625} \][/tex]
[tex]\[ |t| > 6.25 \][/tex]
This inequality tells us that [tex]\( t \)[/tex] can be any value greater than 6.25 seconds or less than -6.25 seconds. Since time [tex]\( t \)[/tex] in this context is usually considered non-negative (Jerald cannot go back in time), we discard the negative interval.
Thus, the interval of time for which 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]
