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Answer :
Sure! Let's solve this problem step-by-step.
We have the equation that models Jerald's height during his bungee jump:
[tex]\[ h = -16t^2 + 729 \][/tex]
where [tex]\( h \)[/tex] is his height in feet and [tex]\( t \)[/tex] is the time in seconds. We need to find out for which interval of time Jerald's height is less than 104 feet.
1. Set Up the Inequality:
We want his height to be less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
2. Solve the Inequality:
First, let's simplify the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Subtract 729 from both sides:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
Which simplifies to:
[tex]\[ -16t^2 < -625 \][/tex]
Divide both sides by -16. Remember, dividing by a negative number flips the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
3. Calculate the Time Intervals:
Now, calculate the square root of [tex]\(\frac{625}{16}\)[/tex]:
[tex]\[ t > \sqrt{39.0625} \][/tex]
Taking the square root gives:
[tex]\[ t > 6.25 \][/tex]
or
[tex]\[ t < -6.25 \][/tex]
However, since time cannot be negative, we discard [tex]\( t < -6.25 \)[/tex].
So, the interval of time during which Jerald is less than 104 feet above the ground is when:
[tex]\[ t > 6.25 \][/tex]
Therefore, the correct answer is:
- [tex]\( t > 6.25 \)[/tex]
I hope this helps! If you have any more questions, feel free to ask.
We have the equation that models Jerald's height during his bungee jump:
[tex]\[ h = -16t^2 + 729 \][/tex]
where [tex]\( h \)[/tex] is his height in feet and [tex]\( t \)[/tex] is the time in seconds. We need to find out for which interval of time Jerald's height is less than 104 feet.
1. Set Up the Inequality:
We want his height to be less than 104 feet:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
2. Solve the Inequality:
First, let's simplify the inequality:
[tex]\[ -16t^2 + 729 < 104 \][/tex]
Subtract 729 from both sides:
[tex]\[ -16t^2 < 104 - 729 \][/tex]
Which simplifies to:
[tex]\[ -16t^2 < -625 \][/tex]
Divide both sides by -16. Remember, dividing by a negative number flips the inequality sign:
[tex]\[ t^2 > \frac{625}{16} \][/tex]
3. Calculate the Time Intervals:
Now, calculate the square root of [tex]\(\frac{625}{16}\)[/tex]:
[tex]\[ t > \sqrt{39.0625} \][/tex]
Taking the square root gives:
[tex]\[ t > 6.25 \][/tex]
or
[tex]\[ t < -6.25 \][/tex]
However, since time cannot be negative, we discard [tex]\( t < -6.25 \)[/tex].
So, the interval of time during which Jerald is less than 104 feet above the ground is when:
[tex]\[ t > 6.25 \][/tex]
Therefore, the correct answer is:
- [tex]\( t > 6.25 \)[/tex]
I hope this helps! If you have any more questions, feel free to ask.
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