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Answer :
Sure, let's solve the problem step by step. We need to find the time interval when Jerald's height is less than 104 feet above the ground.
The equation for Jerald’s height is given by:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when:
[tex]\[ h < 104 \][/tex]
Substitute 104 into the equation:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
To solve this inequality, follow these steps:
1. Subtract 104 from both sides:
[tex]\[-16t^2 + 729 - 104 < 0\][/tex]
Simplify the left side:
[tex]\[-16t^2 + 625 < 0\][/tex]
2. Rearrange the inequality:
[tex]\[-16t^2 < -625 \][/tex]
3. Divide every term by -16:
Since we are dividing by a negative number, we need to flip the inequality sign.
[tex]\[ t^2 > \frac{625}{16} \][/tex]
When you simplify [tex]\(\frac{625}{16}\)[/tex], you get:
[tex]\[ t^2 > 39.0625 \][/tex]
4. Take the square root of both sides:
[tex]\[ |t| > \sqrt{39.0625} \][/tex]
[tex]\(\sqrt{39.0625}\)[/tex] is 6.25, so:
[tex]\[ |t| > 6.25 \][/tex]
This means:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Given that time [tex]\(t\)[/tex] must be a non-negative value in this context (as it represents time elapsed after the jump), we only consider:
[tex]\[ t > 6.25 \][/tex]
Thus, Jerald’s height is less than 104 feet above the ground for the time interval:
[tex]\[ t > 6.25 \][/tex] seconds.
The equation for Jerald’s height is given by:
[tex]\[ h = -16t^2 + 729 \][/tex]
We want to find when:
[tex]\[ h < 104 \][/tex]
Substitute 104 into the equation:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
To solve this inequality, follow these steps:
1. Subtract 104 from both sides:
[tex]\[-16t^2 + 729 - 104 < 0\][/tex]
Simplify the left side:
[tex]\[-16t^2 + 625 < 0\][/tex]
2. Rearrange the inequality:
[tex]\[-16t^2 < -625 \][/tex]
3. Divide every term by -16:
Since we are dividing by a negative number, we need to flip the inequality sign.
[tex]\[ t^2 > \frac{625}{16} \][/tex]
When you simplify [tex]\(\frac{625}{16}\)[/tex], you get:
[tex]\[ t^2 > 39.0625 \][/tex]
4. Take the square root of both sides:
[tex]\[ |t| > \sqrt{39.0625} \][/tex]
[tex]\(\sqrt{39.0625}\)[/tex] is 6.25, so:
[tex]\[ |t| > 6.25 \][/tex]
This means:
[tex]\[ t > 6.25 \quad \text{or} \quad t < -6.25 \][/tex]
Given that time [tex]\(t\)[/tex] must be a non-negative value in this context (as it represents time elapsed after the jump), we only consider:
[tex]\[ t > 6.25 \][/tex]
Thus, Jerald’s height is less than 104 feet above the ground for the time interval:
[tex]\[ t > 6.25 \][/tex] seconds.
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