Answer :
To find the interval of time when Jerald's height is less than 104 feet above the ground, we start with the given equation for his height:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to determine when his height [tex]\( h \)[/tex] is less than 104 feet. So, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, isolate the quadratic term by subtracting 729 from both sides:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Now, divide both sides by -16. Remember that dividing by a negative number reverses the inequality:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Next, calculate the square root of both sides to solve for [tex]\( t \)[/tex]. We are looking for the values of [tex]\( t \)[/tex] that satisfy this inequality:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
The square root of [tex]\(\frac{625}{16}\)[/tex] is [tex]\(\frac{\sqrt{625}}{\sqrt{16}}\)[/tex], which simplifies to [tex]\(\frac{25}{4} = 6.25\)[/tex].
Based on our calculations, we find:
[tex]\[ t > 6.25 \][/tex]
This tells us that Jerald's height is less than 104 feet when the time [tex]\( t \)[/tex] is greater than 6.25 seconds. Therefore, the correct interval is:
[tex]\[ t > 6.25 \][/tex]
Jerald falls below 104 feet after 6.25 seconds.
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to determine when his height [tex]\( h \)[/tex] is less than 104 feet. So, we set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
First, isolate the quadratic term by subtracting 729 from both sides:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
[tex]\[ -16t^2 < -625 \][/tex]
Now, divide both sides by -16. Remember that dividing by a negative number reverses the inequality:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
Next, calculate the square root of both sides to solve for [tex]\( t \)[/tex]. We are looking for the values of [tex]\( t \)[/tex] that satisfy this inequality:
[tex]\[ t > \sqrt{\frac{625}{16}} \][/tex]
The square root of [tex]\(\frac{625}{16}\)[/tex] is [tex]\(\frac{\sqrt{625}}{\sqrt{16}}\)[/tex], which simplifies to [tex]\(\frac{25}{4} = 6.25\)[/tex].
Based on our calculations, we find:
[tex]\[ t > 6.25 \][/tex]
This tells us that Jerald's height is less than 104 feet when the time [tex]\( t \)[/tex] is greater than 6.25 seconds. Therefore, the correct interval is:
[tex]\[ t > 6.25 \][/tex]
Jerald falls below 104 feet after 6.25 seconds.
