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Answer :
To solve this problem, we need to determine when Jerald's height is less than 104 feet. His height as a function of time [tex]\( t \)[/tex] is given by the equation:
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We need to find the values of [tex]\( t \)[/tex] for which [tex]\( h(t) < 104 \)[/tex]. Let's set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
To solve for [tex]\( t \)[/tex], we'll start by isolating the quadratic term:
1. Subtract 104 from both sides to simplify the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Rearrange the terms:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16. Remember, when you divide or multiply an inequality by a negative number, you must reverse the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Simplify the fraction:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides to solve for [tex]\( t \)[/tex]. Since [tex]\( t \)[/tex] represents time and should be non-negative, we only consider the positive root:
[tex]\[ t > \sqrt{39.0625} \][/tex]
6. Calculate the square root:
[tex]\[ t > 6.25 \][/tex]
So, Jerald is less than 104 feet above the ground for the time interval where [tex]\( t > 6.25 \)[/tex].
The correct answer is:
[tex]\( t > 6.25 \)[/tex]
[tex]\[ h(t) = -16t^2 + 729 \][/tex]
We need to find the values of [tex]\( t \)[/tex] for which [tex]\( h(t) < 104 \)[/tex]. Let's set up the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
To solve for [tex]\( t \)[/tex], we'll start by isolating the quadratic term:
1. Subtract 104 from both sides to simplify the inequality:
[tex]\[ -16t^2 + 729 - 104 < 0 \][/tex]
[tex]\[ -16t^2 + 625 < 0 \][/tex]
2. Rearrange the terms:
[tex]\[ -16t^2 < -625 \][/tex]
3. Divide both sides by -16. Remember, when you divide or multiply an inequality by a negative number, you must reverse the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
4. Simplify the fraction:
[tex]\[ t^2 > 39.0625 \][/tex]
5. Take the square root of both sides to solve for [tex]\( t \)[/tex]. Since [tex]\( t \)[/tex] represents time and should be non-negative, we only consider the positive root:
[tex]\[ t > \sqrt{39.0625} \][/tex]
6. Calculate the square root:
[tex]\[ t > 6.25 \][/tex]
So, Jerald is less than 104 feet above the ground for the time interval where [tex]\( t > 6.25 \)[/tex].
The correct answer is:
[tex]\( t > 6.25 \)[/tex]
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