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Answer :
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We are given Jerald's height from the ground as a function of time in the equation [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( t \)[/tex] is the time in seconds. We need to find the interval of time for which Jerald's height is less than 104 feet.
1. Set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the inequality:
Subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
3. Solve for [tex]\( t^2 \)[/tex]:
Divide each side by -16. Remember, dividing by a negative number flips the inequality sign:
[tex]\[
t^2 > \frac{-625}{-16}
\][/tex]
Simplify the division:
[tex]\[
t^2 > 39.0625
\][/tex]
4. Find [tex]\( t \)[/tex]:
To solve for [tex]\( t \)[/tex], take the square root of both sides. The square root of 39.0625 is approximately 6.25:
[tex]\[
t > \sqrt{39.0625}
\][/tex]
[tex]\[
t > 6.25 \text{ or } t < -6.25
\][/tex]
Since time [tex]\( t \)[/tex] must be greater than or equal to 0 (as we cannot have negative time), we only consider the positive solution:
[tex]\[
t > 6.25
\][/tex]
Thus, Jerald is less than 104 feet above the ground when [tex]\( t > 6.25 \)[/tex]. The correct choice that matches our solution is:
[tex]\[
t > 6.25
\][/tex]
We are given Jerald's height from the ground as a function of time in the equation [tex]\( h = -16t^2 + 729 \)[/tex], where [tex]\( t \)[/tex] is the time in seconds. We need to find the interval of time for which Jerald's height is less than 104 feet.
1. Set up the inequality:
[tex]\[
-16t^2 + 729 < 104
\][/tex]
2. Rearrange the inequality:
Subtract 729 from both sides:
[tex]\[
-16t^2 < 104 - 729
\][/tex]
Simplify the right side:
[tex]\[
-16t^2 < -625
\][/tex]
3. Solve for [tex]\( t^2 \)[/tex]:
Divide each side by -16. Remember, dividing by a negative number flips the inequality sign:
[tex]\[
t^2 > \frac{-625}{-16}
\][/tex]
Simplify the division:
[tex]\[
t^2 > 39.0625
\][/tex]
4. Find [tex]\( t \)[/tex]:
To solve for [tex]\( t \)[/tex], take the square root of both sides. The square root of 39.0625 is approximately 6.25:
[tex]\[
t > \sqrt{39.0625}
\][/tex]
[tex]\[
t > 6.25 \text{ or } t < -6.25
\][/tex]
Since time [tex]\( t \)[/tex] must be greater than or equal to 0 (as we cannot have negative time), we only consider the positive solution:
[tex]\[
t > 6.25
\][/tex]
Thus, Jerald is less than 104 feet above the ground when [tex]\( t > 6.25 \)[/tex]. The correct choice that matches our solution is:
[tex]\[
t > 6.25
\][/tex]
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